Adaptive narrowband interference rejection for satellite navigation receiver

ABSTRACT

A controller is configured to control the adaptive notch filter and to execute a search technique (e.g., artificial intelligence (AI) search technique) to converge on filter coefficients and to recursively adjust the filter coefficients of the adaptive notch filter in real time to adaptively adjust one or more filter characteristics (e.g., maximum notch depth or attenuation, bandwidth of notch, or general magnitude versus frequency response of notch).

RELATED APPLICATION

This document (including the drawings) claims priority and the benefitof the filing date based on U.S. provisional application No. 63/093,161,filed Oct. 16, 2020, and entitled ADAPTIVE NARROWBAND INTERFERENCEREJECTION FOR SATELLITE NAVIGATION RECEIVER under 35 U.S.C. § 119 (e),where the provisional application is hereby incorporated by referenceherein.

FIELD OF THE DISCLOSURE

This disclosure document relates to adaptive narrowband interferencerejection for a satellite navigation receiver.

BACKGROUND

The electromagnetic spectrum is limited for wireless communications. Aswireless communications are engineered to support greater datatransmission throughput for end users, the potential for interference tosatellite navigation receivers tend to increase. Interference may becaused by various technical factors, such as inadequate frequencyspacing or spatial separation between wireless transmitters,intermodulation distortion between wireless signals, receiverdesensitization, or deviation from entirely orthogonal encoding ofspread-spectrum signals, outdated radio or microwave frequencypropagation modeling of government regulators, among others.Accordingly, there is need to ameliorate interference through anadaptive narrowband interference rejection system.

SUMMARY

In accordance with one embodiment, a receiver system with interferencerejection, the receiver system comprises an antenna for receiving aradio frequency signal. A downconverter is configured to convert theradio frequency signal to an intermediate frequency signal. Ananalog-to-digital converter is configured to convert the intermediatefrequency signal or an analog baseband signal to a digital basebandsignal. A selective filtering module is arranged to filter or processthe digital baseband signal, where the selective filtering modulecomprises a narrowband rejection filter configured to reject aninterference component that interferes with the received radio frequencysignal. The selective filtering module comprises an adaptive notchfilter that supports an infinite impulse response (IIR). A controller isconfigured to control the adaptive notch filter and to execute a searchtechnique (e.g., artificial intelligence (AI) search technique) toconverge on filter coefficients and to recursively adjust the filtercoefficients of the adaptive notch filter in real time to adaptivelyadjust one or more filter characteristics (e.g., maximum notch depth orattenuation, bandwidth of notch, or general magnitude versus frequencyresponse of notch).

BRIEF DESCRIPTION

FIG. 1A is a block diagram of a receiver system with digital signalprocessing for adaptive narrowband interference rejection for radiofrequency signal, such as a microwave satellite signal.

FIG. 1B is a block diagram a first narrow band rejection system and asecond narrow band rejection system of FIG. 1A in greater detail.

FIG. 2A is a block diagram of an adaptive notch filter using a leastminimum squares (LMS) technique to select, tune and/or adjust filtercoefficients in real time based on a received signal that includesinterference or an interfering signal component.

FIG. 2B is a block diagram of an adaptive notch filter using aSteiglitz-McBride technique (e.g., Steiglitz-McBride interferenceidentification model) to select, tune and/or adjust filter coefficientsin real time based on a received signal that includes interference or aninterfering signal component.

FIG. 3 is a block diagram of a first embodiment of a cascaded notchfilter architecture.

FIG. 4 is a first illustrative example of a magnitude versus frequencyresponse of a notch filter in accordance with the disclosure.

FIG. 5 is a second illustrative example of a magnitude versus frequencyresponse of a notch filter that is associated with various correspondingcontrol parameters (e.g., ρ).

FIG. 6 is a block diagram of a second embodiment of a cascaded notchfilter architecture.

FIG. 7 a block diagram of a third embodiment of a cascaded notch filterarchitecture that illustrates the feedback of the infinite impulseresponse configuration consistent with the Steiglitz-McBride approach.

FIG. 8 is a block diagram of a fourth embodiment of a cascaded notchfilter that comprises a second order variant of the cascaded notchfilter of FIG. 7 .

FIG. 9 is a block diagram of a fifth embodiment of a cascaded notchfilter based on a mini-batch gradient descent algorithm.

FIG. 10 is a theoretical model to predict the signal to noise ratiodegradation associated with an illustrative neighboring bandinterference signal.

FIG. 11 is an illustrative example of a chart of magnitude versusfrequency response of the notch filter, where an adaptive solution andan expected or benchmark solution are compared.

FIG. 12 is an illustrative example of a chart that shows convergenceversus time of the filter coefficients to a steady-state solution offilter coefficients as a reference.

FIG. 13 is an illustrative example of a chart that shows group delay ofthe notch filter over an extended bandwidth of potential interest.

FIG. 14 is an illustrative example of graph of potential filter error ina magnitude of the desired, received signal compared betweenpre-convergence and post-convergence.

FIG. 15 is illustrative example of histogram of the desired, receivedsignal versus an error signal at a steady state.

DETAILED DESCRIPTION

As used in this document, adapted to, arranged to or configured to meansthat one or more data processors, logic devices, digital electroniccircuits, delay lines, or electronic devices are programmed withsoftware instructions to be executed, or are provided with equivalentcircuitry, to perform a task, calculation, estimation, communication, orother function set forth in this document.

An electronic data processor means a microcontroller, microprocessor, anarithmetic logic unit, a Boolean logic circuit, a digital signalprocessor (DSP), a programmable gate array, an application specificintegrated circuit (ASIC), or another electronic data processor forexecuting software instructions, logic, code or modules that arestorable in any data storage device.

As used in this document, a radio frequency signal comprises anyelectromagnetic signal or wireless communication signal in themillimeter frequency bands, microwave frequency bands,ultra-high-frequency bands, or other frequency bands that are used forwireless communications of data, voice, telemetry, navigation signals,and the like.

FIG. 1A is a block diagram of a receiver system 100 with digital signalprocessing for adaptive narrowband interference rejection for radiofrequency signal, such as a microwave satellite signal 102. A globalnavigation satellite system (GNSS) comprises a constellation ofsatellites 101 orbiting around the Earth. Each satellite 101 (e.g., GNSSsatellite) comprises a transmitter for transmitting a desired navigationsatellite signal 102 or radio frequency signal that can be received by areceiver system 100 (e.g., GNSS receiver system or satellite navigationreceiver). Meanwhile, an interfering transmitter 104 may transmit aninterference signal 103 on a frequency and with a modulation that hasthe potential to interfere with the reception of the desired navigationsatellite signal 102 of the GNSS receiver.

The receiver system 100 represents an illustrative example of onepossible reception environment for a radio receiver such as a globalnavigation satellite system (GNSS) receiver. The satellite 101 (e.g.,satellite vehicle) transmits the satellite signal 102 on multiplefrequencies, such that the collective set of signals may be referred toas a composite signal. For example, in FIG. 1A the Lx can represent oneor more satellite signals 102, such as L1, L2, and/or L5 signals, thatare used in a Global Positioning System (GPS). The satellite signal 102will be attenuated or disturbed by free space propagation, theionosphere, and troposphere. In practice, the satellite signal 102 canbe impacted by background noise and/or some potential interferencesignals 103 (e.g., narrowband interference signal). For example, at aterrestrial radio tower, an interfering transmitter 104 may transmit oneor more interference signals 103 within the same or adjacent band to oneor more transmitted satellite signals 102 from the satellite 101. Theinterference signal 103 can be classified as either widebandinterference (WBI, e.g., a pulse-like signal) or narrowband interference(NBI, e.g., a continuous wave (CW) signal, which bandwidth is relativelynarrower than the GNSS signal). The scope of this disclosure willconcentrate on reduction or filtering of interference from one or morenarrowband interference signals 103, although it may be apparent tothose of ordinary skill in the art to adapt the disclosure to widebandinterference.

As used herein, the NBI may be synonymous with an NBI component or NBIcomponents.

FIG. 1A shows a receiver system 100 capable of receiving signalstransmitted by satellites 101 that include one or more carrier signals(e.g., a first carrier (L1), a second carrier (L2) and an additionalthird carrier (L5) of the Global Positioning System (GPS)) such that thereceiver system 100 can determine position, velocity, and attitude(e.g., yaw, tilt and roll angles) with very high accuracy and precisionbased on the received signals. The received signals may be transmittedfrom one or more satellites 101, such as a GPS satellite, aGalileo-compatible satellite, or a Global Navigation Satellite System(GLONASS) satellite. The satellites 101 have approximately known orbitalpositions versus time that can be used to estimate the relative positionbetween an antenna 106 of the receiver system 100 and each satellite101, based on the propagation time of one or more received signalsbetween four or more of the satellites 101 and the antenna 106 of thereceiver system 100.

Precise point positioning (PPP) includes the use of precise satelliteorbit and clock corrections provided wirelessly via correction data,rather than through normal satellite broadcast information (ephemerisand clock data) that is encoded on the received satellite signals, todetermine a relative position or absolute position of a mobile receiver.PPP may use correction data that is applicable to a wide geographicarea. Although the resulting positions can be accurate within a fewcentimeters using state-of-the-art algorithms, conventional precisepoint positioning can have a long convergence time of up to tens ofminutes to stabilize and determine the float or integer ambiguity valuesnecessary to achieve the purported (e.g., advertised) steady-stateaccuracy. Hence, such long convergence time is typically a limitingfactor in the applicability of PPP.

In accordance with one embodiment, FIG. 1A illustrates a receiver system100 with a dual-path receiver configuration. In a dual-path receiverconfiguration with interference rejection, the receiver system 100comprises an antenna 106 for receiving a radio frequency signal, such asa microwave frequency satellite signal (e.g., one or more satellitecarrier signals from multiple satellites, such as at least four orbitingsatellites). The antenna 106 is coupled to a signal splitter 107 thatsplits the received radio frequency signal into a first radio frequencysignal and a second radio frequency signal, where the first radiofrequency signal and the second radio frequency signal are generallyidentical to each other. Further, the first radio frequency signal andthe second radio frequency signal are essentially an attenuated versionof the received radio frequency signal at an output port 105 of theantenna 106. The splitter 107 may comprise a diplexer, a hybrid splitter107, a radio frequency transformer, or the like.

Here, a dual band system is described as an example with a first analogsignal path 156 corresponding to the first radio frequency signal and asecond analog signal path 157 associated with second radio frequencysignal, although in other configurations multiple parallel signal pathsfor corresponding different frequency bands may be used. For example, adual band system includes a low-band and a high band, where the low-bandhas a lower frequency range than the high band does. For the GlobalPositioning System (GPS), the transmitted L1 frequency signal of asatellite 101 may comprise the high band; the transmitted L2 frequencysignal may comprise the low band signal. Further, the L1 carrier is at1,575.42 MHz, which is modulated with the P(Y) code (pseudo random noisecode) and M code that occupies a target reception bandwidth on each sideof the carrier. Meanwhile, the L2 carrier is at 1,227.6 MHz andmodulated with the C/A (coarse acquisition) code, P(Y) code (pseudorandom noise code) and M code that occupies a target reception bandwidthon each side of the carrier. The splitter 107 (e.g., diplexer) splitsthe composite signal into the first signal path (e.g., upper signal pathor high-band path) and the second signal path (e.g., lower path orlow-band path).

In one embodiment, the signal splitter 107 or hybrid may split thereceived signal into two received radio frequency signals for processingby a first analog module 111 and a second analog module 131. Each analogmodule (111, 131) converts the received radio frequency (RF) ormicrowave frequency signal to an intermediate frequency (IF) signal oranalog baseband signal. The first analog module 111 may comprise anoptional first pre-amplifier 141 or a low-noise amplifier (LNA) foramplifying the received signal. Similarly, the second analog module 131may comprise an optional second pre-amplifier 151 or a low-noiseamplifier (LNA) for amplifying the received signal. To simplify thereceiver analog filtering design, the front-end of a typical modern GNSSreceiver uses a wideband front-end design to receive multiple GNSSsignals using two/three wideband filters (not shown), where each bandtargets a target bandwidth (e.g., 140-300 MHz).

In the first analog signal path 156, a first downconverter 142 isconfigured to convert the (amplified) first radio frequency signal to anintermediate frequency signal. For example, the first analog module 111comprises a first downconverter 142, such as the combination of a mixerand a local oscillator, which moves the high band (L1, G1, B1, orsimilar frequency associated with a GNSS) radio frequency (RF) into theintermediate frequency (IF). The first downconverter 142 is coupled tothe first analog-to-digital (ADC) converter 112.

In the first analog signal path 156, a first automatic gain control(AGC) 143 is coupled to the first ADC 112 and the first downconverter142. For example, in one configuration of the first signal path, a firstautomatic gain control (AGC) 143 is coupled to the first ADC 112, thefirst downconverter 142, and the first pre-amplifier 141. The firstautomatic gain control (AGC) 143 may control the gain (e.g., root meansquare (RMS) amplitude) of an input signal to the corresponding firstanalog-to-digital converter (ADC) 112 to be constant or within a targetrange (e.g., despite fluctuations in ambient radio frequency noise andthe interference signal 103). The first AGC 143 receives gain-relatedfeedback from the first ADC 112 to adjust the gain setting of firstdownconverter 142 (and/or the first pre-amplifier 141).

In the second analog signal path 157, a second downconverter 152 isconfigured to convert the (amplified) second radio frequency signal toan intermediate frequency signal. For example, the second analog module131 comprises a second downconverter 152, such as the combination of amixer and a local oscillator, which moves the low band (L2, or similarfrequency associated with a GNSS) radio frequency (RF) into theintermediate frequency (IF).

In the second analog signal path 157, a second automatic gain control(AGC) 153 is coupled to the second analog-to-digital converter (ADC) 132and the second downconverter 152. For example, in one configuration ofthe second signal path, a second automatic gain control (AGC) 153 iscoupled to the second ADC 132, the second downconverter 152, and thesecond pre-amplifier 151. The second automatic gain control (AGC) 153may control the gain (e.g., root mean square (RMS) amplitude) of aninput signal to the corresponding second analog-to-digital converter(ADC) 132 to be constant or within a target range (e.g., despitefluctuations in ambient radio frequency noise and the interferencesignal 103). The second AGC 153 receives gain-related feedback from thesecond ADC 132 to adjust the gain setting of second downconverter 152(and/or the second pre-amplifier 151).

Each analog-to-digital converter (ADC) (112, 132) may be coupled to itscorresponding automatic gain control (AGC) (143, 153) that providesvariable gain amplification. In turn, each AGC is coupled to itscorresponding downconverter (142, 152). In one embodiment, the automaticgain control AGC provides a feedback signal to the downconverter (142,152) or intermediate frequency (IF) filter (e.g., analog IF filter) thatis associated with the downconverter. The downconverter (142, 152) orits analog IF filter adapts the signal voltage (pea-to-peak) within thefirst ADC 112 to be commensurate with its operational range.

In one embodiment, each ADC (112, 132) samples the analog receivedsignal (from the corresponding downconverter (142, 152)) using apredefined sampling rate, which, per Nyquist theorem, should be equal toor greater than two times the bandwidth (e.g., target receptionbandwidth) for the practical sampling design. The bandwidth of the ADCdetermines maximum tolerable interference at a given quantization loss.The resulting digital sequence or filter input (113, 133) reconstructsthe received signal, such as the first signal (e.g., high-band RFsignal) and the second signal (e.g., low-band RF signal) to the basebandsignal with a corresponding baseband bandwidth or range.

In terms of the AGC feedback control from its respective ADC (112, 132),the AGC feedback control can be done either in analog or digital domain.For example, an envelope detector is typically used for AGC and variablegain control if analog control is used. Because of advances in digitalprocessing theory and practice, the digital processing for the AGCfeedback control, can be based on a statistical processes, such asdigital analysis of a histogram of the sample digital stream (or filterinputs 113, 133) at the output of the corresponding analog-to-digitalconverter, ADC (112, 132) to generate a feedback signal to control theAGC (143, 153), such as the first AGC 143 associated with thecorresponding first analog signal path 156 and the second AGC 153associated with the second analog signal path 157. Each AGC is coupledto the downconverter (142, 152), which in practice may comprise thedownconverter and IF filter module with inherent gain/amplificationadjustments.

A first analog-to-digital converter 112 is configured to convert theanalog intermediate frequency signal to a digital intermediate frequencysignal, or to convert an analog baseband signal to a digital basebandsignal. A first selective filtering module 144 is arranged to filter orprocess the digital baseband signal, where the first selective filteringmodule 144 may comprise a first sub-band filter 114 (e.g., pass-bandfilter) and first narrowband rejection system 110 configured to rejectan interference component that interferes with the received radiofrequency signal (e.g., associated with a first sub-band or set ofchannels).

In one embodiment, the selective filtering module (144, 154) comprisesan adaptive notch filter that supports an infinite impulse response(IIR). Within the selective filtering module (144, 154), in oneembodiment an electronic controller or electronic data processor isconfigured to control the adaptive notch filter and to execute a searchtechnique (e.g., artificial intelligence (AI) search technique) toconverge on filter coefficients and to recursively adjust the filtercoefficients of the adaptive notch filter in real time to adaptivelyadjust one or more filter characteristics (e.g., maximum notch depth orattenuation, bandwidth of notch, or general magnitude versus frequencyresponse of notch).

In a first digital signal path 256, a first selective filtering module144 comprises a first narrowband rejection system 110, such as anadaptive notch filter that supports an infinite impulse response (IIR).A controller is configured to control the first adaptive notch filterand to execute a search technique (e.g., artificial intelligence (AI)search technique) to converge on first filter coefficients and torecursively adjust the first filter coefficients of the first adaptivenotch filter in real time to adaptively adjust one or more filtercharacteristics (e.g., maximum notch depth or attenuation, bandwidth ofnotch, or general magnitude versus frequency response of notch).

In a second digital signal path 257, a second analog-to-digitalconverter 132 is configured to convert the analog intermediate frequencysignal to a digital intermediate frequency signal, or to convert ananalog baseband signal to a digital baseband signal. A second selectivefiltering module 154 is arranged to filter or process the digitalbaseband signal, where the second selective filtering module 154 maycomprise a second sub-band filter 134 (e.g., bass-band filter) andsecond narrowband rejection system 130 configured to reject aninterference component that interferes with the received radio frequencysignal (e.g. associated with a second sub-band or a second set ofchannels).

The second selective filtering module 154 comprises a second narrowbandrejection system 130, such as an adaptive notch filter (e.g., one ormore adaptive notch filters) that supports an infinite impulse response(IIR). In the second selective filtering module 154, an electroniccontroller or electronic data processor is configured to control thesecond adaptive notch filter and to execute a search technique (e.g.,artificial intelligence (AI) search technique) to converge on secondfilter coefficients and to recursively adjust the second filtercoefficients of the second adaptive notch filter in real time toadaptively adjust one or more filter characteristics (e.g., maximumnotch depth or attenuation, bandwidth of notch, or general magnitudeversus frequency response of notch).

A selective filtering module, such as a first selective filtering module144 or a second selective filtering module 154, is arranged to filter orprocess the digital baseband signal, where the filtering module maycomprise one or more of the following: (a) a first sub-band filter 114or first channel filter, such as a pass-band filter to filter signalsoutside a target reception bandwidth; (b) a first narrow band rejectionsystem 110, such as a narrowband rejection filter configured to rejectan interference component at a target rejection frequency or within atarget rejection bandwidth that interferes with the received radiofrequency signal; (c) a second sub-band filter 134 or second channelfilter, such as a pass-band filter to filter signals outside a targetreception bandwidth; (d) a second narrowband rejection system 130, suchas a narrowband rejection filter configured to reject an interferencecomponent at a target rejection frequency or within a target rejectionbandwidth that interferes with the received radio frequency signal. Forexample, within each selective filtering module (144, 154) or eachsub-band filter (114, 134), such as a bandpass filter (BPF) attenuatesor rejects the image band of the mixer output of the downconverter (142,152), or frequencies outside of the target reception bandwidth of thereceived signal about its central carrier.

The first selective filtering module 144 comprises a first sub-bandfilter 114 (e.g., digital GNSS band filter) that extracts the targetedcomponent or digital signal from the first signal (e.g., high-bandsignal or whole high band spectrum) at a first node that corresponds tofilter input 113. At the terminal of the output signal 115 of the firstsub-band filter 114, the first resultant signal comprises a GNSS signalat a band of interest (e.g. L1 or G1 or B1), the narrowband interference(NBI), and the noise; the wide-band interference (WBI). Similarly, thesecond selective filtering module 154 comprises a second sub-band filter134 (e.g., digital GNSS band filter) that extracts the targetedcomponent or digital signal from the second signal (e.g., low-bandsignal or whole low band spectrum) at a second node that corresponds tofilter input 133. At the terminal of the output signal 135 of the secondsub-band filter 134, the second resultant signal comprises a GNSS signalat a band of interest (e.g. L2 or G2 or B2), the narrowband interference(NBI), and the noise; the wideband interference (WBI). The WBImitigation is not addressed directly in this disclosure, althoughcertain filtering techniques may have general applicability to both NBIand WBI.

In one example, a relatively strong NBI component in signal 115 willlead to the signal-to-noise ratio (SNR) degradation. In practice, suchSNR degradation is significantly determined by the relative location ofNBI reference to the pseudorandom noise (PN) modulated radio frequencysignal in the frequency domain and the de-spreading gain that a specificPN sequence provides. The quantity analysis of such impact on SNRdegradation or receiver performance will be discussed later in thisdisclosure.

To mitigate the impact of the NBI on the PN sequence demodulationperformance, the first narrowband rejection system 110 (e.g., with oneor more adaptive notch filters) adaptively rejects the NBI. As shown inFIG. 1A, the first narrowband rejection system 110 is coupled to theoutput signal 115 of the first sub-band filter 114. The first narrowbandrejection system 110 filters the first signal to reject NBI in the firstsignal (e.g., in the first digital signal path 256).

Similarly, the second narrowband rejection system 130 (e.g., thatcomprises one or more adaptive filters) filters the second signal toreject NBI in the second signal. As shown in FIG. 1A, the secondnarrowband rejection system 130 is coupled to the terminal of the outputsignal 135 of the second sub-band filter 134. The second narrowbandrejection system 130 filters the second signal to reject NBI in thesecond signal (e.g., in the second digital signal path 257).

At the output of each narrow band rejection system (110, 130), theresidual signal (116, 136), ideally, contains only the PN signal and thenoise because the NBI would be completely eliminated. In practice, theNBI will be attenuated, reduced or ameliorated in the PN signal based onthe performance of the embodiments of the adaptive filtering algorithmsdescribed in this disclosure.

As shown in FIG. 1A, the first selective filtering module 144 comprisesone or more of the following associated with a first digital signal path256: (a) a first sub-band filter 114, such as a first channel filter ora first bandpass filter for filtering a first target reception bandwidth(e.g., GNSS sub-band for L1 or L2 signal), and (b) a first narrow bandrejection system 110, such as first adaptive narrow band interferencerejection filter. The second selective filtering module 154 comprisesone or more of the following associated with a second digital signalpath 257: (a) a second sub-band filter 134, such as a second channelfilter or a second bandpass filter for filtering a second targetreception bandwidth (e.g., GNSS sub-band for L1 or L2 signal), and (b) asecond narrow band rejection system 130, such as a second adaptivenarrow band interference rejection filter.

Equivalently, to its counterpart along the first signal path (e.g., theupper signal path), the second signal path (e.g. the lower signal path)from splitter 107 (e.g., coupler) is processed by the seconddown-converter 152, which comprises an IF filter (e.g., analog IFfilter). The second downconverter 152 is coupled to the second ADC 132.The second AGC 153 is coupled between the second ADC 132 and the seconddownconverter 152 for adjusting the gain or scaling the input signal tothe second ADC 132. At the output of the second ADC 132, the secondresultant digital stream represents the low-band RF signal at basebandrange. The bandpass selective filtering extracts the signal from atargeted band (e.g. L2, L5 etc.). The second NB rejection system 130(e.g., second NBI rejection system) is added to mitigate the PNdemodulation degradation on the targeted band. This disclosure describesillustrative or possible designs (and respective estimated or modeledperformance) to the first narrowband rejection system 110 and the secondnarrowband rejection system 130.

A band-selection multiplexor (MUX) 120 is coupled to the output of thefirst narrowband rejection system 110 (e.g., first adaptive narrow bandrejection filter) and the second narrowband rejection system 130 (e.g.,the second adaptive narrow band rejection filter) to select the digitalsample stream or channel that represents the first signal or the secondsignal, where each signal may be modulated with or encoded with targetPN sequence or another encoding scheme. For example, the band-selectionmultiplexer selects the first signal, the first channel or L1 band ifthe targeted PN sequence type is GPS L1 CA.

The appropriate sample stream will be further processed by the GNSSchannel processing module 145, which typically comprises: one ormultiple carrier phase demodulators, a replica or local PN codegenerator sampled at multiple delayed phases, banks of correlators, andmultiple accumulators, to create a bank of in-phase (I) and quadrature(Q) measurements at an interval of millisecond (ms) or multiplemilliseconds to drive the baseband tracking loop.

Further, in some embodiments the GNSS channel processing module 145 mayfurther comprise a binary offset sub-carrier (BOC) modulator (used formodern GNSS signal such as GPS L1C, BeiDou B1C, Galileo E1 signalsetc.).

In one embodiment, the GNSS channel processing module 145 may comprise abaseband tracking loop module for tracking of code phase and carrierphase. For example, the baseband tracking loop module derives correctionor control signals to control a local oscillator or the numericallycontrolled oscillators (NCOs) in the GNSS signal processing module tomaintain the synchronization between the received signal in the channeland the local replica of that channel with respect to code phase andcarrier phase. A GNSS processing module 145 is coupled to a navigationprocessing module 155.

In one embodiment, the navigation processing module 155 takes apseudo-range measurements and carrier phase measurements and otherrelated information from the satellites 101 to generate the positioningsolution, which is used as a feedback to align the receivercrystal-grade clock (e.g., of lower temporal precision) with thesatellite-based atomic grade clock (e.g., of higher temporal precision);the solution also, combined with other information, generates thein-view satellites 101 list to control the appropriate receiver resourceallocation.

FIG. 1B is a block diagram a first narrowband rejection system 110 and asecond narrowband rejection system 130 of FIG. 1A in greater detail.

In FIG. 1B, the first narrowband rejection system 110 comprises a firstelectronic data processor 160, a first data ports 161, and first datastorage device 163 coupled to a first data bus 162. The first electronicdata processor 160, the first data ports 161 and first data storagedevice 163 may communicate data messages with each other via the firstdata bus 162, for example. In one embodiment, the first data storagedevice 163, may store filter parameters, filter coefficients, referencefilter parameters, reference filter coefficients, and softwareinstructions relative to adaptive filters, predictive filters, leastminimum squares (LMS) search algorithms for filter parameters and filtercoefficients, minimum mean square error (MMSE) search algorithms forfilter parameters and filter coefficients, Steiglitz-McBride models forestimating or determining filter parameters and filter coefficients, andmodified Steiglitz-McBride models for estimating or determining filterparameters and filter coefficients. As illustrated, the first datastorage device 163 may store software instructions or software modulerelated to a primary adaptive notch filter (200, 220, 300 or 600) and asecondary adaptive notch filter (700, 800 or 900), such as primary firstadaptive notch filter and a secondary first adaptive notch filter,respectively.

In alternate embodiments, alone or cumulative with the above primaryfirst adaptive notch filter and the secondary first adaptive notchfilter, the first data storage device 163 may store program instructionsfor any of the following: (a) filter emulation; (b) initializing,estimating, updating, resetting, storing and retrieving filtercoefficients and filter parameters; (c) configuring, controlling,communicating operating digital circuitry (e.g., digital signalprocessor coupled to the first data ports 161) comprising delay lines,summers, shift registers, adders, and other digital components.

In FIG. 1B, the second narrow band rejection system 130 comprises asecond electronic data processor 170, a second data ports 171, andsecond data storage device 173 coupled to a second data bus 172. Thesecond electronic data processor 170, the second data ports 171 andsecond data storage device 173 may communicate data messages with eachother via the second data bus 172, for example.

In one embodiment, the second data storage device 173, may store filterparameters, filter coefficients, reference filter parameters, referencefilter coefficients, and software instructions relative to adaptivefilters, predictive filters, least minimum squares (LMS) searchalgorithms for filter parameters and filter coefficients, minimum meansquare error (MMSE) search algorithms for filter parameters and filtercoefficients, Steiglitz-McBride models for estimating or determiningfilter parameters and filter coefficients, and modifiedSteiglitz-McBride models for estimating or determining filter parametersand filter coefficients. As illustrated, the second data storage device173 may store software instructions or software module related to aprimary adaptive notch filter (200, 220, 300 or 600) and a secondaryadaptive notch filter (700, 800 or 900), such as a primary secondadaptive notch filter and a secondary second adaptive notch filter,respectively.

In alternate embodiments, alone or cumulative with the above primarysecond adaptive notch filter and the secondary second adaptive notchfilter, the second data storage device 173 may store programinstructions for any of the following: (a) filter emulation; (b)initializing, estimating, updating, resetting, storing and retrievingfilter coefficients and filter parameters; (c) configuring, controlling,communicating operating digital circuitry (e.g., digital signalprocessor coupled to the second data ports 171) comprising delay lines,summers, shift registers, adders, and other digital components.

Within the second data storage device 173, filter parameters, filterparameters and filter emulation, delay lines, summers, shift registers,adders, and other structures may be configured. As illustrated, the datastorage device comprises a primary second adaptive notch filter and asecondary second adaptive notch filter.

In FIG. 1B, the first electronic data processor, the second electronicdata processor, or both may comprise one or more electronic dataprocessors. Each electronic data processor comprises one or more of thefollowing: a microprocessor, a multi-core microprocessor, amicrocontroller, a programmable logic device, a programmable gate array,an arithmetic logic unit, a Boolean logic unit, an electronic logiccircuit or system, a digital circuit, a digital signal processor (DSP),and application specific integrated circuit (ASIC) or another dataprocessing device. In one embodiment, the electronic data processor canexecute software instructions stored in the data storage device 46. Forexample, the electronic data processor can execute software instructionsto facilitate, support, incorporate, call, configure or emulate any ofthe following: digital delay lines, shift registers, memory registers,memory stacks, summers, adders, digital filter components, a digitalnotch filter, predictive or adaptive filter modules, filter parameters,filter coefficients, and artificial intelligence (AI) based control ofadaptive filters.

In FIG. 1B, the first data storage device, the second data storagedevice or both may comprise one or more data storage devices. Each datastorage device may comprise one or more of the following: electronicmemory, nonvolatile electronic memory, shift registers, memory stacks,delay lines, registers, nonvolatile random access memory, a magneticstorage device, an optical storage device, or any other device forstoring and retrieving digital data and/or analog data.

The first data ports 161 and second data ports 171 may comprise one ormore of the following data ports or input/output data ports. Each dataport (161, 171) may comprise a buffer memory and an electronictransceiver for communicating data messages to a network element or viaa communications network, such as the Internet or a wirelesscommunications network (e.g., cellular phone network, or high-bandwidthsmartphone data communications wireless network).

In one embodiment, in the first digital signal path 256, the first dataports 161 may support the receipt and data processing of digitalbaseband signals from the first ADC 112 in the first selective filteringmodule 144. Meanwhile, in the second digital signal path 257, the seconddata ports 171 may support the receipt and data processing of digitalbaseband signals from the second ADC 132 in the second selectivefiltering module 154.

In alternate embodiments, one or more nearby particular receiver systems100 (e.g., GNSS receivers) may be associated with first wirelesscommunication devices that transmit or share filter coefficients and/orgeographical locations associated with NBI at that nearby particularGNSS receivers that are associated with second wireless communicationdevices, where the wireless communication devices are coupled to thefirst data ports, the second data ports, or both.

In FIG. 2A, a block diagram represents a mathematical model of a primaryadaptive notch filter 200, such as an adaptive NBI filter or adaptiveNBI rejection system that comprises an adaptive notch filter 201 incommunication with an adaptive control module 202 that is configured toexecute software instructions or an algorithm, such as a least meansquare (LMS) algorithm, or alternately or cumulatively, aSteiglitz-McBride (SM) algorithm, or a quasi-Steiglitz-McBride (QSM)algorithm. For example, in a least-mean-square (LMS) mode the adaptivecontrol module 202 or electronic data processor (160, 170) searches forsuitable filter coefficients for the adaptive notch filter that minimizethe error in the desired frequency response or notch. For example, theLMS mode supports searching candidate filter coefficients until theadaptive control module 202 converges an optimal or final solution of asuitable, final or optimal filter coefficient that minimizes error inthe frequency response or notch for a corresponding epoch of the GNSSsystem or a sampling time interval associated with the receivedsatellite signal. The adaptive notch filter 201 may comprise a finiteimpulse response filter, or an infinite impulse response filter, or ahybrid filter that serially or sequentially (e.g., in any order) appliesa finite impulse response filter and an infinite impulse responsefilter, to realize the notch filter to reject or attenuate a narrow bandinterference component. The adaptive notch filter 201 typically receivesan input signal from a first sub-band filter 114 (e.g., first bandpassfilter) of the first selective filter module 144 or the second sub-bandfilter 134 (e.g., second bandpass filter) of the second selective filtermodule 154. The adaptive control module 202 can sample or evaluate theoutput of the adaptive notch filter 201 to adjust or determine theoptimal, final or suitable filter coefficients for the adaptive notchfilter 201 for a corresponding GNSS epoch or sampling interval of thereceived signal (e.g., received GNSS satellite signal).

As used throughout this document, an epoch refers to a measurement timeperiod or interval of a global navigation satellite system (e.g., GNSS)receiver, such as a Global Positioning System (GPS) receiver. Forexample, an epoch may be associated with one or more sampling intervalsof carrier phase measurements of a GNSS receiver 100 or its navigationprocessing module 155; or corresponding states of a predictive filter,like a Kalman filter, for estimating position, velocity or attitude ofthe GNSS receiver system 100 (e.g., GNSS receiver). An epoch can bereferenced to a universal time or universal satellite-based time, with atemporal offset.

As used in this document a quasi-Steiglitz-McBride (QSM) algorithm,module, process or mode may represent one or more of the followingalgorithms, modules, processes or modes, together with Steiglitz-McBridealgorithm, module, process or mode: (a) a least mean square (LMS)algorithm module, processor or mode; (b) a minimum-mean-square-error(MMSE) module, processor or mode; and (c) a hybrid or dual modecomprising the Steiglitz-McBride (SM) algorithm, module, process or modeand the least mean square (LMS) algorithm module, processor or mode, orthe minimum-mean-square-error (MMSE) module, processor or mode, or allthree. For example, the QSM mode may support operation of the SMestimator in the SM mode if the convergence period after initializationis lesser than a threshold time period (e.g., to achieve convergence ofthe filter coefficients via a regression process applicable to theinfinite impulse response filter model of the adaptive notch filter).Alternately, the QSM mode may support operation of the SM estimatoroutside of the SM mode, such as within an LMS mode or MMSE mode, if theconvergence period after initialization is lesser than a threshold timeperiod (e.g., to achieve convergence of the filter coefficients via aregression process applicable to the infinite impulse response filtermodel of the adaptive notch filter).

In FIG. 2A, the adaptive notch filter 201 is associated with filtercoefficients that establish a target frequency response that rejectsNBI. For example, the adaptive notch filter 201 is associated withfilter coefficients that establish a notch (e.g., an attenuation regionor attenuation versus frequency response) in a frequency spectrum of thereceived NBI (e.g., for any given GNSS epoch), where the notch or signalattenuation is associated with a generally narrow frequency range of theNBI or a notch center frequency. The notch may be measured in decibelsof attenuation relative to the desired signal or target received signal,for example.

In one embodiment in FIG. 2A in the LMS mode, the primary adaptive notchfilter 200 (e.g., adaptive notch filter (ANF)) comprises an adaptivecontrol module 202 that uses a least mean square (LMS) algorithm toadjust recursively or iteratively the filter coefficients of theadaptive notch filter 201 (e.g., adaptive notch filter module). Forexample, in the LMS mode the suitable, optimal or final filtercoefficients are adjusted to minimize error in the frequency response ofthe adaptive notch filter 201 such that the steady-state (SS) frequencyresponse of adaptive notch filter 201 attenuates the power at NBIfrequencies while passing information over other frequencies.

In FIG. 1A, FIG. 1B, or FIG. 2A, the adaptive notch filter 201, thefirst narrowband rejection system 110, or the second narrowbandrejection system 130 can be configured as a finite impulse response(FIR) or an infinite impulse response (IIR). The IIR filter typicallyhas feedback from its output to drive the selection of filtercoefficients, which can rapidly provide a sharp filter response withminimal data processing of the data processor. The FIR type model can beregarded as a special case of the IIR architecture discussed in thisdisclosure, but the FIR filter is usually more computationallyintensive. Moreover, compared with the FIR architecture, the IIRarchitecture can provide a pre-emphasis filtering to enhance temporarilythe NBI for detection purposes, which improves the NBI estimationembedded in the noise. The temporary enhancement may be referred to asan adaptive line enhancer (ALE) and is embedded in the adaptive notchfilter. Accordingly, the IIR filer configuration with he ALE facilitatesimproved estimation of NBI components to improve prediction of thefilter coefficients; hence, minimizes the residual interferencecomponent in the filter output signal 203 (e.g., output samples), orfilter output of FIG. 2A.

In one embodiment, the adaptive control module may further comprise apredictive module. A complex IIR-base adaptive notch filter (ANF)algorithm can use a Gauss-Newton type algorithm to align or synchronizethe notch frequency or ANF indentation frequency with the NBI. In steadystate (SS) mode the predictive module or electronic data processor (160,170) evaluates the stored signals sampled at the previous epoch of thereceiver system 100 to accurately predict the NBI component, such as theNBI center frequency or NBI bandwidth, of the received signal of thereceiver system 100 sampled at current epoch. Accordingly, thepredictive module can drive the selection of filter coefficients thatminimize error in the notch filter response compared to the NBIcomponent with its corresponding predicted NBI center frequency orcorresponding predicted NBI bandwidth. Further, in one implementation,the digital adaptive notch filter (e.g., 200, 220, 300, 600, 700, 800,or 900) can simply subtract the prediction from the current sample torealize the notch. After the subtraction, the resultant signal containsonly the pseudo-random noise (PN) signal, the uncompensated residual ofNBI, and noise (e.g., associated with background noise, white noise, orthe noise floor).

To the extent that the adaptive notch filter 201 is configured as an IIRfilter, the IIR filter can be susceptible to stability risks. Theadaptive control module 202 may be configured to use theSteiglitz-McBride (SM) model, alone or in conjunction with thepredictive filter module. The adaptive control module 202 may operate inthe LM mode, or the SM mode, or a hybrid mode that includes both the LMmode or the SM mode for certain corresponding GNSS epochs, for example.

If the adaptive control module operates 202 in the SM mode, it is wellsuited for promoting the efficiency and convergence stability of the IIRconfiguration of the adaptive notch filter. The SM algorithm introducesan iterative solution, per the criteria of minimum-mean-square-error(MMSE), rather the LMS. The SM IIR model is associated with solvingoptimal regression equations that are highly nonlinear and intractable.However, a de-correlated two path model can be used convert theestimation of an auto-regression (AR) model into a combination of twomoving average (MA) models, among other things. The equations set forthin this disclosure facilitate the technical details to solve efficientlythe regression equations in the context of an adaptive notch filterbased on the SM model and IIR adaptive filter configuration.

In FIG. 2A, the adaptive notch filter 201 receives an input signal(e.g., 115, 135 in FIG. 1A) which in this illustrative example maycomprise a carrier modulated by a PN sequence, the receiving systemnoise, and the NBI. The adaptive notch filter 201 (e.g., adaptive notchfilter module) can be incorporated into the first narrowband rejectionsystem 110 and the second narrowband rejection system 130, or both.

In one example, the adaptive notch filter 201 may be based on theautoregressive-moving-average (ARMA) or moving average (MA) model,either of which can create attenuation at NBI center frequency tomitigate the NBI impact on the PN sequence demodulation. The ARMA modelprovides two polynomials to model a process including: (1) anautoregression polynomial and (2) a moving average (MA) polynomial. Theadaptive notch filter 201 can adapt its notch frequency per the NBIcomponent in signal (e.g., 115, 135) to minimize the NBI component.Regardless of whether the adaptive filter control module 202, uses theMA or ARMA model to control the adaptive notch filter 201, the conceptis to minimize the total power (e.g., hence, minimizing total noisepower indirectly) of the filter output signal 203; hence, mitigate oreliminate an NBI.

The adaptive control module 202 receives a filter output signal 203 ofthe adaptive notch filter 201 to derive an error signal that can be usedby the least mean square (LMS) or recursive least square (RLS) algorithmto adjust the filter coefficients of the adaptive filter 201 (e.g.,adaptive notch filter module). That is, the output signal includes anerror signal component or quasi-noise component. In the steady-state(SS) mode, the error signal will be noise like. Therefore, for the SSmode, the mean adjustment by the adaptive control module 202 is “zero”and the filter coefficients of adaptive notch filter have converged to astable state. Equation 1 provides a unified model to both FIR-basedadaptive notch filter (ANF) and IIR-based ANF as follows:

$\begin{matrix}{{{H(z)} = \frac{A(z)}{A\left( {\rho\; z} \right)}},} & {{Equation}\mspace{14mu} 1}\end{matrix}$where H(z) is a Z-transform of the filter transfer function, where A(z)defines the frequency (or frequencies) to be attenuated or target notchfrequency (or frequencies) (e.g., frequency response versus magnitude)of the adaptive notch filter, where A(ρz) also defines the frequency orfrequencies (e.g., frequency response versus magnitude) to be attenuated(e.g., notched) and that are scaled by filter parameter ρ (e.g.,contributory filter shape factor or bandwidth control factor).

For example, the filter parameter, ρ, applies to a IIR filter if ρ≠0.Although a Z-transform is used other frequency domain representationscould be used; where Fourier transforms are more typically used formodeling FIR filters. In practice, the adaptive notch filter 201comprises a series of delay lines, adders, summers, data storageregisters, shift registers, or accumulators (e.g., realized inprogrammable logic arrays, digital circuitry or digital signalprocessors) that are accessible via taps to apply filter coefficients orfilter parameters to digital samples of the received signal to defineadjust the frequency versus magnitude response of the adjustable filterto reduce or eliminate narrowband interference.

In one embodiment, the filter parameter ρ and the filter coefficientβ_(i), collectively, define filter shape factors of the frequency versusmagnitude response for the adaptive notch filter 201 (e.g., IIR notchfilter) that can affect the notch bandwidth and attenuation magnitude ofdepth of the notch of the adaptive notch filter. If ρ≠0, the denominatorA(ρz), in Equation 1, cannot be reduced to 1; therefore, the resultantmodel is an IIR adaptive notch filter. However, if ρ=0, we obtainA(ρz)=1, i.e. the Equation 1 degenerates into a FIR based ANF; the“zeros” of the numerator A(z) defines the frequency to be attenuated ortarget notch frequency (or frequencies) of the adaptive notch filter. Tobetter illustrate the concept, Equation 2 represents A(z) using amultiplication format as follows:

$\begin{matrix}{{{A(z)} = {{\prod_{i = 1}^{I}\left( {1 - {\beta_{i}e^{j\omega_{i}}z^{- 1}}} \right)} = {1 + {\sum_{i = 1}^{I}{a_{i}^{*}z^{- i}}}}}},} & {{Equation}\mspace{14mu} 2}\end{matrix}$where

-   -   A(z) defines the frequency (or frequencies) to be attenuated,        such as a target notch frequency or target notch frequencies of        the adaptive notch filter;    -   Π_(i=1) ^(I)(1−β_(i)e^(jω) ^(i) z⁻¹) is a multiplication        representation of a set of NBI components from a first NBI        component to an ith NBI component up to a total of I NBI        components;    -   I indicates the total number of the narrow band interference        components (e.g., NBIs) that need to be rejected or notched, or        the order of the adaptive notch filter or the degree of freedom        (D.O.F);    -   e^(jω) ^(i) is the ith NBI component modeled as a simple        continuous wave (CW);    -   β_(i) is a filter coefficient of the adaptive notch filter that        determines a notch depth to the ith NBI and in range of [0, 1];    -   1+Σ_(i=1) ^(I) a*_(i)z^(−i) is a sum representation, equivalent        to the multiplication representation Π_(i=1) ^(I)(1−β_(i)e^(jω)        ^(i) z⁻¹), where a*_(i) is the conjugate complex to a_(i);    -   a_(i) is a filter coefficient of the adaptive notch filter.        For example, with respect to Equation 2, β_(i)=0 means there is        no attenuation applied to the ith NBI component; while β_(i)=1        means total or perfect elimination of ith NBI component as        1−β_(i)e^(jω) ^(i) z⁻¹=0. The sum representation makes it easy        to execute the partial derivative therefore widely used for the        system identification. For Equation 2, Newton's identities        establish the relationship between the {β_(i)e^(jω) ^(i)        }|_(i=1 . . . I) and {a_(i)}|_(i=1 . . . I),        Equation 3 applies the adaptive notch filter 201 (e.g., IIR        adaptive notch filter) as follows:

$\begin{matrix}{{{{A\left( {\rho z} \right)} = {\prod_{i = 1}^{I}\left( {1 - {\rho\beta_{i}e^{j\omega_{i}}z^{- 1}}} \right)}}{= {1 + {\sum_{i = 1}^{I}{a_{i}^{*}\rho^{i}z^{- i}}}}}},} & {{Equation}\mspace{14mu} 3}\end{matrix}$where

-   -   A(ρz) defines the frequency (or frequencies) to be attenuated,        such as a target notch frequency or target notch frequencies        (e.g., target magnitude versus frequency response) of the        adaptive notch filter;    -   ρ is a filter parameter, which in combination with filter        coefficient β_(i) can be used to control the bandwidth and depth        of the notch to each narrowband interference component (NBI),    -   Π_(i=1) ^(I)(1−β_(i)e^(jω) ^(i) z⁻¹) is a multiplication        representation of a set of NBI component from a first NBI        component to an ith NBI component up to a total of I NBI        components;    -   I indicates the total number of the narrow band interference        (NBI component) that need to be rejected or notched, or the        order of the adaptive notch filter or the degree of freedom        (D.O.F);    -   e^(jω) ^(i) is the ith NBI component modeled as a simple        continuous wave (CW);    -   β_(i) is a filter coefficient of the adaptive notch filter that        determines a notch depth to the ith NBI component and in range        of [0, 1];    -   z^(−i) represents a time delay or delay line (e.g., in the        z-transform domain) associated with the ith tap of the adaptive        narrow band filter;    -   1+Σ_(i=1) ^(I) a*_(i)z⁻¹ is a sum representation, equivalent to        the multiplication representation Π_(i=1) ^(I)(1−β_(i)e^(jω)        ^(i) z⁻¹), where a*_(i) is the conjugate complex to a_(i);    -   a_(i) is a filter coefficient of the adaptive notch filter; and    -   1+Σ_(i=1) ^(I) a*_(i)ρ^(i)z^(−i) is the sum representation of        denominator; this term cannot be reduced for the IIR based        adaptive notch filter.        The adaptive notch filter 201, such as a FIR based adaptive        notch filter (ρ=0), can be realized by sampling the frequency        response of the narrow band interference component of the        received signal and taking an inverse Fourier transform, or by        using a least minimum squares (LMS) method to estimate filter        coefficients. Similarly, the FIR based adaptive notch filter may        estimate filter coefficients in accordance with a chi-square        minimization approach, such as minimum mean square error (MMSE)        method, that assumes a Gaussian error distribution. However,        this disclosure focuses on a practical design of the IIR based        on the adaptive notch filter, where in the above equations:        (ρ≠0).

FIG. 2B is a realization of the system of FIG. 2A that applies theSteiglitz-McBride (SM) identification model to provide an error signalto the adaptive control module 202 to update or iteratively convergeupon the filter coefficients of the adaptive notch filter 201. Asillustrated in FIG. 2B, the SM technique creates two signal processingpaths (e.g., in the z-transform domain) for comparison. The first signalprocessing path 275 (e.g., upper path in FIG. 2B) filters the receivedsignal sampled at epoch k−d (future epoch of the GNSS) using the filtercoefficients of the adaptive filter sampled at epoch k−d; the secondsignal processing path 276 (e.g., lower path) filters the receivedsignal sampled at epoch k−d using the filter coefficients of theadaptive filter sampled at epoch k−1 (e.g., previous epoch of the GNSS);the difference between the samples, accumulators or registers at theoutput of digital processing signal paths provides the error signal tothe adaptive control module 202 as an input to update the filtercoefficients.

In FIG. 2B, in the first signal processing path 275 (e.g., the uppersignal processing path) the first adaptive line enhancement (ALE) module222 uses the filter coefficients estimated at epoch k−1 (e.g., previousepoch) and enhances/increases the magnitude of frequency componentsclose to the NBI components. In contrast, in the first signal processingpath 275 the inverse adaptive line enhancement (ALE) module 225 appliesan inverse ALE computation to estimate the filter coefficients at epochk (following the previous epoch) or future epoch k−d, which attenuatesthe frequency components of the NBI components, where the inverse ALEcomputation may be scaled appropriately (e.g., in magnitude) to bothcounteract or offset the first ALE module 222 and to attenuate or notchthe target NBI component.

In FIG. 2B in the steady-state (SS) mode, where D_(k)(z)˜=D_(k−1)(z),the output signal 226 of the first signal processing path 275 isapproximately equal to the input signal (e.g., 115, 135) sampled atepoch k. Along the second signal processing path 276 (e.g., the lowersignal path), the received signal sampled at epoch k, passes through adelay module 51 to delay the signal by d epochs, where d can be anyknown or arbitrary number of epochs. The delayed signal 233 is appliedto or passes through the second ALE module 232 based on the coefficientsestimated at epoch k−1.

In the SS mode, the resultant ALE output 234 of the second ALE module232 is processed by the narrowband interference (NBI) estimator 235. TheNBI estimator 235 estimates one or more NBI components embedded in thereceived signal (e.g., based on input signals 115, 135, or both) sampledat epoch k.

Therefore, in SS mode the filter output signal 203 of FIG. 2B isapproximately equal to the input signal (e.g., 115, 135). Outside of theSS mode, the filter output signal 203 comprises an error signal wherethe output signal is determined by subtraction at summer 250 of the NBIestimation signal 236 (e.g., NBI digital samples) sampled at epoch k andthe output signal 226 of the inverse ALE module 225. Further, outsidethe SS mode, the filter output signal 203 comprises residual NBIcomponents, the desired PN signal itself, and the noise. The errorsignal can be defined as the residual NBI component, for example.

In FIG. 2B, the mathematical model of the narrowband interferenceestimator 235, by applying the Equation 2 and Equation 3 is as follows:

$\begin{matrix}{{{{\left\lbrack {{A\left( {\rho z} \right)} - {A(z)}} \right\rbrack z^{d}} = {\left\lbrack {1 + {\sum_{i = 1}^{I}{a_{i}^{*}z^{- i}}} - 1 - {\sum_{i = 1}^{I}{a_{i}^{*}\rho^{i}z^{- i}}}} \right\rbrack z^{d}}}{= {\left\lbrack {\sum_{i = 1}^{I}{{a_{i}^{*}\left( {1 - \rho^{i}} \right)}z^{- i}}} \right\rbrack z^{d}}}},} & {{Equation}\mspace{14mu} 4}\end{matrix}$where

-   -   z^(d) indicates a time advance unit (e.g., in the z-transform        domain);    -   z^(−i) represents a time delay or delay line (e.g., in the        z-transform domain) associated with the ith tap of the adaptive        narrow band filter;    -   ρ is a filter parameter (e.g., contributory notch shaping        factor), where ρ^(i) is the filter parameter associated with the        ith narrowband interference;    -   a_(i) is a filter coefficient of the adaptive notch filter,        where a*_(i) is the conjugate complex to a_(i); and    -   I indicates the total number of the narrow band interference        (NBIs) that need to be rejected or notched, or the order of the        adaptive notch filter or the degree of freedom (D.O.F);        With respect to z^(d), if the input signal (to the NBI estimator        235) is sampled at epoch k, the output signal (of clock for the        NBI estimator 235) is sampled at future epoch k+d. However, any        system (e.g., system 220 of FIG. 2B) is not physically        realizable if it requires future information, such as        information about a future epoch. Because the i in the Equation        4 starts from 1, [Σ_(i=1) ^(I) a*_(i)(1−ρ^(i))z^(−i)] introduces        a delay of one sample, the digital data processing block of the        NBI estimator 235 is physically realizable for only d=1, without        further refinements to be discussed later in this disclosure. To        achieve the optimal performance, the noise term of the signal        (115, 135) sampled at epoch k through the first signal        processing path 275 should be independent from the delayed noise        term in delayed signal 233 processed by the second signal        processing path 276. As illustrated in FIG. 2B, the first signal        processing path 275 comprises the first ALE module 222 and the        inverse ALE module 225, where the first ALE module 222 is        coupled to the inverter ALE module 222 at terminal(s) or node        223. Meanwhile, the second signal processing path 276 comprises        the delay module 51, the second adaptive line enhancement (ALE)        module 232, and the NBI estimator 235. The over-sampling system        commonly requires d>1 to decorrelate the noise terms sampled at        different epochs.

The Gauss-Newton adaptive algorithm used to adjust the adjustable,adaptive notch filter 201, or namely, the filter coefficients vector,which will be is described later in this disclosure. The input andoutput relationship can be defined by the following vector relationshipassuming Equation 5:

$\begin{matrix}{\overset{\rightharpoonup}{\theta} = \left\lbrack {a_{1},a_{2},\ldots\mspace{14mu},a_{I}} \right\rbrack^{T}} & {{Equation}\mspace{14mu} 5}\end{matrix}$After the summation (e.g., subtraction) of the data samples from thefirst data processing path (branch) and the second data processing path(branch), the (notch) filter output signal 203 can be written asfollows:

$\begin{matrix}\begin{matrix}{{E(k)} = {{R(k)} + {\sum\limits_{i = 1}^{I}{a_{i}^{*}\left\lbrack {{R\left( {k - 1} \right)} - {\rho^{i}{E\left( {k - i} \right)}}} \right\rbrack}}}} \\{= {{R(k)} + {{\overset{\rightharpoonup}{\theta}}^{H}{\varphi(k)}}}}\end{matrix} & {{Equation}\mspace{14mu} 6}\end{matrix}$where

-   -   E(k) denotes the filter output signal 203 or output error signal        sampled at epoch k,    -   R(k) denotes the input signal or received signal (115, 135)        sampled at epoch k,    -   a*_(i) denotes the conjugate to the ith element, or the        conjugate of a_(i) which is the ith filter coefficient (e.g.,        associated with the ith tap of the adaptive notch filter), of        estimated vector        ,    -   ^(H) represents the conjugate transpose of vector        ,    -   φ(k)={R(k−i)−ρ^(i)E(k−i)}|_(i=1 . . . I) is the vector of        estimation sampled from epoch k−I to k−1.

One possible goal of the primary adaptive notch filter 200 (e.g.,adaptive notch filter system) of FIG. 2A or the primary adaptive notchfilter 220 (e.g., adaptive notch filter system) of FIG. 2B is toeliminate a narrowband interference component (e.g., a narrowbandinterference that has a signal strength or magnitude that exceeds athreshold signal strength) in the input signal (e.g., 115, 135).Therefore, the energy of the filter output signal 203, compared with theenergy of signal (e.g., 115, 135), should be significantly reducedbecause of the attenuation associated with the adaptive notch filter201. Here, the gradient-descent method is used to find the global minimaon the energy surface of E(k), the output signal (or output errorsignal) that includes an error component, spanned by I-parameter vector{right arrow over (θ)}. The negative gradient-descent vector is definedas

$\begin{matrix}{{\nabla_{R}{E(k)}} = {\left\lbrack {\frac{\partial}{\partial a_{1,R}},\ \frac{\partial}{\partial a_{2,R}},\ldots\mspace{14mu},\frac{\partial}{\partial a_{I,R}}} \right\rbrack^{T}{E(k)}}} & {{Equation}\mspace{14mu} 7} \\{{{\nabla_{I}{E(k)}} = {\left\lbrack {\frac{\partial}{\partial a_{1,I}},\ \frac{\partial}{\partial a_{2,I}},\ldots\mspace{14mu},\ \frac{\partial}{\partial a_{I,I}}} \right\rbrack^{T}{E(k)}}},} & {{Equation}\mspace{14mu} 8}\end{matrix}$where

-   -   ∀_(R) is the partial derivative to the real part of filter        coefficient a or the set of filter coefficients a_(i) which        comprise the ith filter coefficient from i equals 1 to I equals        I,    -   ∀_(I) is the partial derivative to the imaginary part of filter        coefficient a or the set of filter coefficients a_(i) which        comprise the ith filter coefficient from i equals 1 to I equals        I.

$\begin{matrix}{{{\psi(k)} = {{- \frac{1}{2}}\left( {{\nabla_{R}{+ j}}\nabla_{I}} \right){E(k)}}},} & {{Equation}\mspace{14mu} 9}\end{matrix}$which by applying Equation 6, becomes Equation 10.

$\begin{matrix}{{{\psi(k)} = {{\varphi(k)}/{A\left( {\rho\; z} \right)}}},} & {{Equation}\mspace{14mu} 10}\end{matrix}$where

-   -   φ(k)={R(k−i)−ρ^(i)E(k−i)}|_(i=1 . . . I) is the vector of        estimation sampled from epoch k−I to k−1; and    -   A(ρz) defines the frequency (or frequencies) to be attenuated,        such as a target notch frequency or target notch frequencies        (e.g., target magnitude versus frequency response) of the        adaptive notch filter; and    -   ψ(k) is a function (or gradient descent function) of the        derivatives of the filter coefficients and the output signal        E(k) for any epoch k of the GNSS (e.g., output signal of the        block diagram of FIG. 2B that includes an error component in the        output signal).

Referring to FIG. 3 , the above gradient descent function ψ(k) isconsistent with an adaptive notch filter (e.g., 300) that has a cascadedIIR structure. The above illustration of the gradient descent derivationwill be referenced in later to clarify the cascaded adaptive notchfilter structure 300, which can increase the attenuation on theinterference signal without deteriorating the stability and adding thecomplexity of the algorithm.

FIG. 3 is a physically feasible digital data processing modelarchitecture of an adaptive notch filter 300 using the Steiglitz-McBride(SM) method or quasi-Steiglitz-McBride (QSM) method. The adaptive notchfilter 300 of FIG. 3 is similar to the adaptive notch filter 220 of FIG.2B, except the notch filter 300 of FIG. 3 further comprises a pre-scaler331, first delay module 231 and secondary notch module 333 (e.g., postprocessing block in FIG. 3 ). The first delay module 231 and the seconddelay module 251 are similar to the delay module 51 of FIG. 2B. Likereference numbers in FIG. 2A, FIG. 2B and FIG. 3 indicate like elements,features, processes or modules. Compared with the adaptive notch filter220 outlined in FIG. 2B, the cascaded, adaptive notch filter 300 of FIG.3 is well-suited for high speed (e.g., 40 MHz or faster) multi-bitsystem data processing.

In the block diagram of the adaptive notch filter 300 of FIG. 3 , thepre-scaler 331 (e.g., pre-scale block) is used to reduce the dynamicrange of the input signal (115, 135) of the received signal or channel,which is used to compensate the gain amplification of the first ALEmodule 222 and the second ALE module 232. Each ALE module (222, 232) hasa gain, G_(ALE), that is described by Equation 11.

$\begin{matrix}\begin{matrix}{G_{ALE} = {\prod\limits_{i = 1}^{I}{1\text{/}{{1 - {\rho \cdot e^{j\;{\Delta\omega}_{i}}}}}}}} \\{{= {\prod\limits_{i = 1}^{I}{1\text{/}\sqrt{1 - {2{\rho \cdot {\cos\left( {\Delta\omega}_{i} \right)}}} + \rho^{2}}}}},}\end{matrix} & {{Equation}\mspace{14mu} 11}\end{matrix}$where

-   -   ρ is the bandwidth control factor or contributory notch shape        factor introduced in Equation 3.    -   Δω_(i)=ω_(i)−ω is the difference between the angular frequency        (in radians) of ith interference and the averaged angular        frequency (e.g., geometric mean, a moving average, or a weighted        average based on the more recent sampling intervals) over I        interferences,    -   ∥ is the absolute value norm of a complex number.        For example, the gain of the first ALE module 222, the second        ALE module 232, or both, for the first order notch filter (I=1)        is 1/√{square root over (1−2ρ·cos(Δω₁)+ρ²)}≤1/(1−ρ). The        reduction of the dynamic range of the adaptive filter 300        significantly saves the logic usage in the practical        implementation as the processing is widely involved with        multi-bits complex number calculation.

In FIG. 3 , the time advance unit z^(d) (previously in the NBI estimator235 of FIG. 2B on the second signal processing path 275) moves to thefirst signal processing path (375 of FIG. 3 ) as first delay module 231.At a reference tap the signal 304 or delayed sample is outputted by thefirst delay module 231. In FIG. 3 , such equivalent time shift betweenthe first signal processing path 375 and the second signal processingpath 376 makes the notch adaptive filter 300 of FIG. 3 physicallyrealizable for any delay of d (d>=1). The cascaded adaptive notch filter300 can be designed to be slightly over-sampled to guarantee thesatisfaction of Nyquist criteria. Accordingly, compliance with theNyquist criteria or reduction of quantization noise tends to require d>1to decorrelate the noise term in the input signal (115, 135) sampled atepoch k and the delayed input signal 313 sampled at epoch k−d. Becausethe first delay module 231 of the first signal processing path 375 ofFIG. 3 and the second delay module 251 of the second signal processingpath 376 of FIG. 3 , the first delay module 231 and the second delaymodule 251, individually and collectively, support physical realizationof any arbitrary delay of d along the second signal processing path 376against the first signal processing path 375. The first signalprocessing path 375 may be referred to as the first data processing pathand the second signal processing path 376 may be referred to as thesecond data processing path.

The secondary notch module 333 (e.g., post-processing block) isconfigured to increase the depth of notch (or magnitude of attenuation)without adding to the degree of freedom (DOF). To illustrate theimportance of the notch depth, as a reference the notch depth and 3-dBbandwidth for the first order notch filter is defined later in thisdocument. Although the first order notch filter is a simple structure,it does provide useful insight to understand the notch filterperformance versus the tunable parameters in the model.

For a first order notch filter, FIG. 4 illustrates the maximum notchattenuation and bandwidth versus the magnitude of absolute value ofnorm-1 filter coefficient |a₁|; various drawings in the disclosureprovide the visual illustration of notch filter behavior versus thebandwidth control parameter ρ. To improve the notch depth, thedisclosure describes a direct implementation of a cascaded notch filtersystem of FIG. 3 that features a secondary notch module 333.

FIG. 6 , FIG. 7 , FIG. 8 and FIG. 9 disclose various embodiments of thecascaded adaptive filter or secondary adaptive notch filter (600, 700,800, and 900) where one or more the embodiments may replace or bepreferred to first order notch filters in accordance with theirrespective characteristics and features disclosed herein that areconsidered with respect to amelioration of one or more NBI components.

By applying Equation 2 and Equation 3 into Equation 1 and using thefirst order notch filter (I=1) as an illustration, the notch depth ofthe filter can be defined by the following Equation 12:

$\begin{matrix}{{D = \frac{\left. {1 +} \middle| a_{1} \middle| {}_{2}{- 2} \middle| a_{1} \middle| {co{s\left( {\Delta\omega_{1}} \right)}} \right.}{\left. {1 - {2\rho}} \middle| a_{1} \middle| {{\cdot {\cos\left( {\Delta\omega}_{1} \right)}} + \rho^{2}} \middle| a_{1} \right|^{2}}},} & {{Equation}\mspace{14mu} 12}\end{matrix}$where

-   -   Δω₁ is the difference of the angle frequency between the        receiver estimation and the true transmission;    -   |a₁| is the norm-1 magnitude of the filter coefficient a₁;    -   |a₁|² is the norm-2 magnitude of the coefficient a₁; and    -   ρ is the bandwidth control factor or contributory notch shape        factor introduced in Equation 3.        In one example, the norm-1 is the vector norm for a vector that        is defined by a matrix form with complex filter coefficients        (e.g., with real and imaginary components) from an ith to I        complex filter coefficient, where I equals 1 for a first order        filter. In another example, the norm-1 represents a maximum        column sum of a matrix of filter coefficients (e.g., from an ith        to I complex filter coefficient), whereas the norm-2 may        represent a maximum of an eigen value of dot product of filter        coefficient matrix and a transpose of the filter coefficient        matrix (e.g., from an ith to I complex filter coefficient).

Based on the above Equation 12, 3 dB bandwidth of the above notch filtercan be determined by

$\begin{matrix}\begin{matrix}{{BW}_{3{dB}} = {{2{\Delta\omega}\mspace{11mu}{{st}.{D({\Delta\omega})}}} = {1\text{/}2}}} \\{{= {2 \cdot \cos^{{- 1}\frac{1 + {{({2 - \rho^{2}})}{a_{1}}^{2}}}{{({4 - {2\rho}})}{a_{1}}}}}},}\end{matrix} & {{Equation}\mspace{14mu} 13}\end{matrix}$where

-   -   BW_(3dB) denotes the 3 dB bandwidth of notch filter;    -   cos⁻¹ is the inverse function of cos( );    -   |a₁| is the norm-1 magnitude of the filter coefficient a₁;        |a₁|² is the norm-2 magnitude of the coefficient a₁; and    -   ρ is the bandwidth control factor or contributory notch shape        factor introduced in Equation 3.

FIG. 4 is a first illustrative example of a magnitude 401 (e.g.,relative magnitude) versus frequency 403 response of a primary adaptivenotch filter (e.g., 200, 220, 300) in accordance with the disclosure.FIG. 4 shows the magnitude 401 (e.g., in Decibels (dB)) along thevertical axis, whereas the frequency 403 (e.g., normalized frequency) isalong the horizontal axis. In particular, FIG. 4 provides a visualillustration regarding the |₁| impact on the notch depth and bandwidth(e.g., half-power bandwidth or 3 dB bandwidth) of the primary adaptivenotch filter (e.g., 200, 220, 300). The interference rejection frequency405 (e.g., typically aligned with an interference frequency by design)is assumed at 0.2 (of the normalized frequency), the real frequency canbe obtained by multiplying with the sampling frequency, F_(s), in Hertz(Hz). The point 381 denotes the 3 dB degradation or attenuation on theleft side of the notch, equal to ω₁−Δω₁; and the point 382 denotes the 3dB degradation or attenuation on the right side of the notch, equal toω₁+Δω₁. Table 1 lists the results shown by FIG. 4 , which demonstratesthat the notch depth increases with the increase of norm-1 magnitude ofthe filter coefficient a₁ |a1| of the primary adaptive notch filter(e.g., 200, 220, 300).

TABLE 1 Notch Depth and 3 dB Bandwidth vs. |a₁| Curve Ref. MaxAttenuation |a₁| No. Line Style or Notch Depth 3 dB BW 0.7 383 Short   −1.9 0 dashes 0.8 384 Alternating    −3 0.025 short and long dashes0.9 385 Long    −5.6 0.045 dashes 1 386 Solid >−10 0.035

In FIG. 4 , a first notch response 383 of the magnitude 401 of signalstrength (e.g., attenuation) versus normalized frequency 403 isindicated by short dashes that define a first curve; a second notchresponse 384 is indicated by alternating long and short dashes thatdefine a second curve; a third notch response 385 is indicated by longdashes that define a third curve; a fourth notch response 386 isindicated as solid line that defines a fourth curve with the greatestattenuation at the rejection frequency 405.

FIG. 5 is a second illustrative example of a magnitude 401 versusfrequency 403 response of a notch filter that is associated with variouscorresponding control parameters p. Like reference numbers in FIG. 4 andFIG. 5 indicate like features or elements, for example.

FIG. 5 provides a visual illustration of notch filter bandwidth (e.g.,of primary adaptive notch filter 200, 220, 300) and maximum attenuation(e.g., notch depth) versus the bandwidth control parameter ρ. Table 2lists the corresponding results shown by FIG. 5 . With the ρ approaches1, the notch bandwidth of the primary adaptive notch filter (e.g., 200,220, 300) becomes narrower and the maximum notch attenuation is reduced.When the ρ=1, from the Equation 1, the denominator A(ρz) reduces toA(z), making the whole notch filter system reduces to H(z)=1, anall-pass filter. Therefore, to design a meaningful notch filter, thebandwidth control parameter ρ can never be 1. Also, the design needs tobalance the target bandwidth against the precision bits (or size of thefixed-point computation or floating number) in the electronic dataprocessor (e.g., 160, 170) or set of electronic data processors. Thecloser that p approaches to 1, the more bit is required to differentiateA(ρz) from the A(z). Here, Table 2 suggests a sufficient bandwidthcontrol parameter may range from approximately 0.7 to approximately 0.9,although other values of the bandwidth control parameter ρ may fallwithin the scope of the claims.

TABLE 2 Notch Depth and 3 dB Bandwidth vs. ρ Curve Ref. Max AttenuationP No. Line Style or Notch Depth 3 dB BW 0.7 391 Short dash −30 0.115 0.8392 Alternating −28 0.075 short and long dash 0.9 393 Long dash −220.035 1 394 Solid    0 0

In FIG. 5 , a first notch response 391 of the magnitude 401 of signalstrength (e.g., attenuation) versus normalized frequency 403 isindicated by short dashes that define a first curve; a second notchresponse 392 is indicated by alternating long and short dashes thatdefine a second curve; a third notch response 393 is indicated by longdashes that define a third curve; a fourth notch response 394 isindicated as a solid line, flat response, or unattenuated referencemagnitude that has a generally constant magnitude 401 of signal strengthversus frequency 403; hence, does not attenuate the input signal at thefrequency of a narrowband interference signal or rejection frequency401.

Equation 12 defines the attenuation can be obtained at Δω₁ offset fromthe true interference frequency. The basic primary adaptive notch filter(e.g., 200, 220) as Equation 1 can be cascaded (e.g., as notch filter300) to create a multiple stage adaptive notch filter consistent withEquation 14 as follows:

$\begin{matrix}{{{H_{N}\left( e^{j\omega_{1}} \right)} = {H^{N}\left( e^{j\omega_{1}} \right)}},} & {{Equation}\mspace{14mu} 14}\end{matrix}$where

-   -   N is the number of stages; and    -   e^(jω) ^(i) is the ith NBI modeled as a simple continuous wave        (CW), where for the first order filter model i and I equals 1.        Compared with single stage notch filter H(z), the N-stage        cascaded system can improve the attenuation or maximum notch        depth, in terms of decibels dB, by up to a factor of N, which is        the number of stages of the cascaded notch filter.

FIG. 6 is a block diagram of a second embodiment of a cascaded notchfilter 600 and its architecture. In particular, FIG. 6 illustrates adirect implementation of H_(N)(z) of Equation 12, where the filtermodules/blocks of the notch filter 600 of FIG. 6 resemble the filtermodules/blocks of the notch filter 300 of FIG. 3 . Like referencenumbers in FIG. 3 and FIG. 6 indicate like elements, features orprocesses.

In FIG. 6 , the pre-scaler 331 is coupled to the filter data processingsystem 602 to adjust the gain inputted to the adaptive notch filter 600(e.g., filter data processing system), such as one or more adaptive lineenhancers (e.g., 622, 632). In FIG. 6 within the filter data processingsystem 602, the first signal path 475 comprises the first ALE (e.g.,adaptive line enhancer) module 622 that is coupled to the first delaymodule 231. In turn, the first delay module 231 is coupled to an inverseALE module 625. The second signal path 575 comprises the second delaymodule 251 that is coupled to the second ALE module 632. In turn, thesecond ALE module 632 is coupled to the narrowband estimator 650.

A summer 250 sums the output signal/samples 306 (in FIG. 6 ) of theinverse ALE module 625 with the output signal/samples 316 of thenarrowband interference estimator to yield an error signal that feedsthe optional cascaded, secondary notch module 601. The optional natureof the cascaded secondary notch module 601 is indicated by the dashedlines in FIG. 6 .

Separately or cumulatively with the optional cascaded secondary notchmodule 601, the notch filter 600 of FIG. 6 may comprise an N cascadedsystem of modules within the first signal path 475 and the second signalpath 476. The cascaded system of modules in the notch filter 600 can bemodeled, at least partially, based on the modules within the firstsignal path 375 and the second signal processing path 376, respectively,of notch filter 300 of FIG. 3 . To illustrate the potential data storagerequirements and computational resources required to implementembodiment of FIG. 6 , N=2 can be used in conjunction with some of theprevious equations:

$\begin{matrix}{{H_{2}(z)} = \frac{A^{2}(z)}{A^{2}\left( {pz} \right)}} & {{Equation}\mspace{14mu} 15}\end{matrix}$Applying Equation 2 and Equation 3 into Equation 13, the numerator ofEquation 13 is set forth in Equation 16.

$\begin{matrix}{{A^{2}(z)} = {1 + {\sum_{l \neq m}{a_{l}^{*}a_{m}^{*}z^{- {({l + m})}}}} + {\sum_{i}{\left( a_{i}^{2} \right)^{*}z^{{- 2}i}}}}} & {{Equation}\mspace{14mu} 16}\end{matrix}$Correspondingly, the resulting error signal (versus Equation 6) becomes

$\begin{matrix}{{E_{2}(k)} = {{R(k)} + {\sum\limits_{lm}^{\neq}{a_{l}^{*}a_{m}^{*}\left\{ {{R\left( {k - l - m} \right)} - {\rho^{2}{E\left( {k - l - m} \right)}}} \right\}}} + {\sum_{i}{\left( a_{i}^{2} \right)^{*}\left\{ {{R\left( {k - {2i}} \right)} - {\rho^{2}{E\left( {k - {2i}} \right)}}} \right\}}}}} & {{Equation}\mspace{14mu} 17}\end{matrix}$Similar to Equation 5, the new parameter vector to Equation 15 can bewritten in Equation 18 as follows:

$\begin{matrix}{\overset{\rightharpoonup}{\theta_{2}} = \left\lbrack {\left( {a_{1}^{2},{a_{1}a_{2}},\ldots\mspace{14mu},{a_{1}a_{I}}} \right),\ldots\mspace{14mu},\left( {{a_{I}a_{1}},{a_{I}a_{2}},\ldots\mspace{14mu},a_{I}^{2}} \right)} \right\rbrack^{T}} & {{Equation}\mspace{14mu} 18}\end{matrix}$Compared with {right arrow over (θ)} of Equation 5, {right arrow over(θ₂)}, although there are still I independent parameters the requisitepotential data storage and data computational resources appear to haveincreased because of one or more of the following reasons: the size is Itimes larger than the basic notch filter system H(z);

-   -   1. each element in the vector is a multiplication of two        independent elements, i.e. the E₂(k) is nonlinearly dependent on        the set fad {a_(i)}|_(=1 . . . I);    -   2. the delayed electronic memory needs to store        {R(k−i)}|_(i=1 . . . I) and {E(k−i)}|_(i=1 . . . 2I) where i        from 1 to 2I, which doubles the electronic memory usage compared        with the basic notch filter system H(z);    -   3. From the derivation of the gradient descent on vector        described by Equation 7 and Equation 8, the fact 1) increases        the complexity of the derivative of        by I times; the fact 1) also degrades the reliability of the        linear feedback system; the fact 2) implies that elements of        I²-length vector        are dependent and are derived from the base set        {a_(i)}|_(i=1 . . . I); the fact 2) shows the nonlinearity        between the base parameter set fad {a_(i)}|_(i=1 . . . I) and        the output error term, making each derivative much more        complicated than a linear term.

Accordingly, for the notch filter 600 of FIG. 6 , the N=2 examplesupports the conclusion that the complexity of the direct method isproportional to I^(N), where I is the order of the basic notch filtersystem and N is the number of cascaded stages.

FIG. 7 a block diagram of a third embodiment of a cascaded notch filteror secondary notch filter 700 that illustrates the feedback of theinfinite impulse response configuration consistent with theSteiglitz-McBride (SM) approach or QSM approach. Like reference numbersin FIG. 2A, FIG. 2B, FIG. 3 and FIG. 6 indicate like elements, featuresand processes.

In FIG. 7 , the secondary notch module 333 (e.g., secondary notch filtermodule or cascaded notch filter block), of the secondary adaptive notchfilter 700, consistent with the SM technique or QSM technique, isintroduced to obtain the deeper attenuation in the magnitude of thenotch depth. Further, the cascaded notch filter of FIG. 7 with the SMtechnique or QSM facilitates one or more of the following items (e.g.,for data storage efficiency and/or computational resource efficiency ofthe one or more data processors):

1) Eliminating the dependency between the elements in the estimatedparameter,

2) Reducing the dimension of

by I^(N-1) times,

3) Reducing the electronic memory size by (N−1)×I times,

4) And obtaining the greater or deeper attenuation using the cascadedsystem.

In the notch filter 300 of FIG. 3 , the set of parameter {right arrowover (θ)} with DOF of I at epoch k−1 are used to determine the first ALEmodule 222 and the second ALE module 232, the updated set of parameter{right arrow over (θ)} with DOF of I at epoch k are used to determinethe inverse ALE module, the NBI estimator 245 and the secondary notchmodule 333.

The secondary adaptive notch filter 700 of FIG. 7 is a detailedillustration of the realization of an Ith order basic notch filter withN-cascaded stages, which can be derived from the basic notch filter 300of FIG. 3 . In FIG. 7 , the filter coefficients vector (e.g.,Theta_(k−1)) 456 sampled at epoch k−1 determines the first ALE (by thefirst ALE module 302) and the second ALE (by the second ALE module 312).The first data processing path 465 delays the ALE output 303 (of thefirst ALE module 302) by d, thus creating the output g(k−d) sampled atepoch k−d based on the filter coefficients at k−d−1. The second dataprocessing path 467 generates the ALE output 314 (of the second ALEmodule 312) based on the delayed received signal sampled at k−d.Therefore, the output from second ALE module 312 h(k−d) is also sampledat epoch k−d based on the filter coefficients at k−1. The second ALEmodule 312 appears in FIG. 7 , FIG. 8 and FIG. 9 .

In FIG. 7 , FIG. 8 and FIG. 9 , the secondary adaptive notch filter(700, 800, 900) embodies an inverse ALE module (e.g., 225) as a firstdelay line assembly 410 and embodies a NBI estimator (e.g., 235 or 245)as a second delay line assembly 440. Each delay line assembly consistsof I delay line units, corresponding to the order of notch filter of I,from i equals 1 to I. In one configuration, each delay line unitrepresents a delay of one sample. In the first delay line assembly 410,the first delay line 361 outputs or stores the first delayed ALE outputof first tap 404; the second delay line 361 outputs or stores the seconddelayed ALE output of a second tap 406; the third delay line 361 (or Ithdelay line) outputs or stores the third delayed ALE output of a thirdtap 409. For example, the first delayed ALE output of a second tap 406may be notated as a delay of g(k−d−1); the second delayed ALE output ofa second tap 406 may be notated as a delay of g(k−d−2); and the thirddelayed ALE output of a third tap 409 may be notated as a delayg(k−d−3); the Ith delayed ALE output may be notated as a delay ofg(k−d−I). In the second delay line assembly 440, the first delay line363 outputs or stores the first delayed ALE output 435; the second delayline 363 outputs or stores the second delayed ALE output of a second tap436; the third delay line 363 (or Ith delay line) outputs or stores thethird delayed ALE output of a third tap 439. For example, the firstdelayed ALE output 435 may be notated as a delay of h(k−d−1); the seconddelayed ALE output of a second tap 436 may be notated as a delay ofh(k−d−2); and the third delayed ALE output of a third tap 439 may benotated as a delay h(k−d−3); the Ith delayed ALE output may be notatedas a delay of h(k−d−I).

As used in this document, the reference to any filter coefficient vectorindicates any of the following: (a) the input signal and the outputsignal (of the filter or adaptive filter) that is (or are) filtered maybe expressed as one or more vectors (e.g., with real and imaginarysignal components, or with phase and magnitude signal components); (b)the input signal and the output signal (of the filter or adaptivefilter) may be expressed as one or more data samples that can bearranged in a matrix or multidimensional array; (c) filter coefficientsmay apply to a digital filter (e.g., of any filter order) such as aninfinite impulse response (IIR) filter, a finite impulse response filter(FIR) or a combination, cascade or hybrid of IIR or FIR, or any adaptivefilter set forth in this document; and (d) filter coefficients (e.g.,scalar values typically referred to as a, b, or a₁, a₂ . . . a_(N) andb₁, b₂ . . . b_(N), where N is any positive integer greater than 2 thatdepends on filter order) related to a transfer function or input/outputfunction, which can be expressed mathematically in the Z-transformdomain, as a Laplace transform (e.g., S-transform), or in anotherfrequency domain representation.

In FIG. 7 , FIG. 8 , and FIG. 9 in the first data processing path (e.g.,branch)(365, 465, 565, respectively), each delayed signal is scaled bythe first set of filter factors (filter coefficient vector 456 in FIG. 7; first filter coefficient estimate 505, second filter coefficientestimate 506 in FIG. 8 ; filter coefficient vector 656 in FIG. 9 ) whichare geometrically proportional to the delay. For example, after thefirst delay line 361 (e.g., z⁻¹) at the corresponding first tap 404, thefirst delayed data sample 415 (e.g., first designated signal) is equalto the input signal 304 (or delayed digital samples at the correspondingreference tap 414) scaled by a*₁ρ¹, where the input signal 304 isinputted to the respective inverse ALE module (225, 625) or delay lineassembly 410. After the second delay line 361 (e.g., z⁻¹) at thecorresponding second tap 406, the second delayed data sample 416 (e.g.,designated signal) is equal to the input signal, of a first tap 404, (ordelayed digital samples) scaled by a*₁ρ², and so on until the Ith delayline and associated Ith tap is scaled by) scaled by a*_(I)ρ^(I).

Meanwhile, in the second data processing path (e.g.,branch)(367,467,567), each delayed signal line is scaled by the secondset of filter factors (filter coefficients vector 456 in FIG. 7 ; firstfilter coefficient estimate 505, second filter coefficient estimate 506in FIG. 8 ; 656 in FIG. 9 ) which are geometrically proportional to thedelay. For example, after the first delay line 363 (e.g., (z−1)) at thefirst tap 435, the designated signal 445 (or delayed digital samples) isequal to the input signal 314 (or delayed digital samples (e.g., from asecond ALE module output)) scaled by a*₁(1−ρ¹); after the second delayline 363 (e.g., (z−1)) at the second tap 436, the second delayed datasample 416 (e.g., designated signal) is equal to input signal, at afirst tap 435, scaled by a*₂(1−ρ²). After the third delay line 363(e.g., (z⁻¹)) at the corresponding third tap 439, the signal 449 (orsecond delayed data sample) is scaled specifically by a*₃(1−ρ³), orgenerally by a*_(I)(1−ρ^(I)), and so on until the Ith delay line andassociated Ith tap.

The signals (or data samples) from the first delay line assembly 410 (inthe respective first data processing path 365, 465, 565) are paired withthe signals (or data samples) from the second delay line assembly 440(in the respective second data processing path 367, 467, 567), wherethere are a primary pair of first delayed data samples (415, 445) and asecond pair of second delayed data samples (416, 446). Each pair ofsignals or data samples is summed by a respective summer 250 to generatea corresponding combined signal (424, 425, 426, 429, 439), which can beinputted to adder 502 via virtual, logical or physical communicationpaths, such as communication paths. For example, the first delay line361 (e.g., first delay unit) and the first delay line 363 (e.g., anotherdelay unit) provide (e.g., simultaneously) a primary pair of firstdelayed data samples (415, 445) to create the signal 425 (e.g., firstsummer output) by summer 250. Similarly, the second delay line 361(e.g., second delay unit) and the second delay line 363 (e.g., anotherdelay unit) provide (e.g., simultaneously) the secondary pair of seconddelayed data samples (416, 446) creates the signal 426 (e.g., secondsummer output) by summer 250. Further, an Nth delay line 361 (e.g.,third delay unit) and the Nth delay line 363 (e.g., another delay unit)provide (e.g., simultaneously) an Nth pair of data samples (419, 439),such as third delayed data samples (419, 439) to create the signal 429(e.g., third summer output) by summer 250. In one embodiment, the adder502 sums the I combined signals from I delay units (361, 363) to resultin the filter output signal 603 with an error signal component. In oneembodiment, within the adder 502 the aggregate data samples from thesummers 250 are accumulated in a buffer memory or registers for adding.

The configurations of the secondary adaptive notch filters (700, 800,900) of FIG. 7 , FIG. 8 and FIG. 9 , respectively, each use feedback tofacilitate determination of the filter coefficients or filtercoefficients and filter parameters. The illustrated examples of FIGS. 7,8 and 9 use IIR filter configurations with feedback. With respect tosuch feedback in FIG. 7 and FIG. 9 , the combined signals or summedsignals from the I delayed units, such as signals 425, 426, . . . , and429, define a vector signal 453 (Ph_(Ik)) which is used to update bothcovariance matrix (e.g., by the covariance update module 457) andcoefficients vector (e.g., by the coefficients update module 458) forthe secondary adaptive notch filter (700, 900).

In FIG. 7 , the covariance update module 457 takes the vector signal 453and the covariance matrix P sampled at epoch k−1 to generate thecovariance matrix 455 (e.g., update module output data) sampled at epochk (e.g., P_(k)). Cumulatively, the coefficients update module 458 takesthe filter coefficients vector 456 sampled at epoch k−1 (e.g., P_(K)−1),the updated covariance matrix 455 sampled at epoch k, and the filteroutput signal 603 (e.g., error signal) sampled at epoch k to generatethe updated filter coefficients vector 459 (e.g., Theta_(k)) sampled atepoch k. In FIG. 7 , the vector signal 459 (e.g., Theta_(k) orTheta_(k+1)) can be applied to the first data processing path 365 andthe second data processing path 367 as feedback by first ALE module(e.g., 302) and second ALE module (e.g., 312) for the epoch k+1, or aspart of a cascaded feedback system of the secondary adaptive notchfilter 700. Similarly, in FIG. 9 , the vector signal (e.g., Theta_(k)(659) or Theta_(k−q) (656)) can be applied to the first data processingpath 565 and the second data processing path 567 as feedback to firstALE module 302 and second ALE module 312 for the epoch k+1, or as partof a cascaded feedback system for the secondary adaptive notch filter900.

The secondary adaptive notch filter 700 of FIG. 7 comprises a feedbacksystem or subsystem is involved with the I×I covariance matrix and I×1vector calculation. For example, in FIG. 7 , both modules 457 and 458require or can use an I×I matrix multiplication, I×1 vectormultiplication, and normalization division. Similarly, in FIG. 9 , boththe covariance update module 657 and the coefficients update module 658require or can use an I×I matrix multiplication, I×1 vectormultiplication, and normalization division. In the practical applicationspecific integrated circuit (ASIC) implementation, the multi-bit matrixmultiplication is a challenge for the high-speed system. In addition,the division can be as difficult as any division algorithm, such asGauss-Newton gradient descent or radix-2 approach. Accordingly, thealgorithm may require multiple steps to converge, for example to asystem using radix-2 divisional algorithm, to satisfy the feedbacksystem or feedback subsystem to operate at a sampling frequency Fs(e.g., within a range from approximately 40 MHz to approximately 300MHz). Further, in FIG. 7 the covariance update module 457 and thecoefficients update module 458 generally needs to operate at M×Fs, whereM is the bit width of the dividend. This challenge can tend to make thefeedback system or subsystem infeasible to fit for some high-speedprocessing systems.

To tackle the aforementioned challenges and to make the adaptive notchfilter and its association feedback system suitable for the high-speedmulti-bit application, this disclosure can simplify the algorithm byapplying some reasonable limitations as set forth in the simplified(e.g., reduced complexity adaptive notch filter system 800 of FIG. 8 .In order to eliminate the matrix and vector multiplication, as shown inFIG. 8 , the simplified, cascaded adaptive notch filter 800 reduces thedelay units of the first signal processing path 565 and of the secondsignal processing path 567 to only two delay lines (361, 363) (i.e. I=2)for each signal processing path (565, 567) and such cascaded adaptivenotch filter is able to attenuate up to two NBI components. Because thismodel of FIG. 8 reduces to the second order notch filter, the filtercoefficients vector consists of two elements, such as parameter vector(e.g., 425, 426), or first filter coefficient estimate 505 (e.g., filterparameter elements), second filter coefficient estimate 506 (e.g.,filter parameter elements), 506 and 546.

With respect to FIG. 8 , the feedback module will be emphasized becausethe rest of the modules are the same as those in notch filter of FIG. 7, where like elements in any two figures in this document indicate likeelements, features or processes. In FIG. 8 , for the feedback system,the bit width of the combined signals (435, 426) can be relatively widebecause of the multiplication calculation along the first signalprocessing path 465 and the second delay signal processing path 467. InFIG. 8 , the first regulator 515 and second regulator 516 can be modeledas sign function (or alternately a sine function or cosine function),which can ignore, round, approximate, or estimate the magnitudeinformation and round or approximate the phase information to the centerof each quadrant among the four states. The resultant signals, firstregulated output 525 and second regulated output 526, only have fourpossible states of complex values: {1+i, 1−i, −1+i, −1−i}, such asquadrature type modulation states. The first regulator and secondregulator (515, 516) significantly reduce the complexity of calculationin the first update module 535 and second update module 536. As also,the signals 525 and 526 are determined or processed separately by firstregulator 515 and second regulator 516, implying the correlation, whereelements of the covariance matrix are not located on the diagonalposition, between the two coefficients can be neglected as well tosimplify matrix calculations by the data processor (160, 170) or theApplication Specific Integrated Circuit (ASIC) or other anotherelectronic data processor.

In FIG. 8 , to tackle the challenge of multi-bit division in thehigh-speed processing mode, the first norm approximation module 512introduces a magnitude approximation. The magnitude of the complexsignal 332 (e.g., digital data samples inputted to the first NBrejection system 110 and the second NB rejection system 130) is round upto the closest exponent of 2 which is modelled by:

$\begin{matrix}{{A_{approx} = 2^{\lceil{\log_{2}A}\rceil}},} & {{Equation}\mspace{14mu} 19}\end{matrix}$where

-   -   A is the magnitude of complex signal 332,    -   ┌ ┐ is the ceiling function,    -   A_(approx) is the resultant approximation to the magnitude.        To the extent that Log₂ in the Equation 19 is still not feasible        for any specific data processor (160, 170) or ASIC        implementation, it be restated as equivalent or approximated to        the following Equation 20:

$\begin{matrix}{{A_{approx} = 2^{{\underset{i}{argmax}{\{{b_{i} = 1}\}}} + 1}},} & {{Equation}\mspace{14mu} 20}\end{matrix}$where

$\underset{i}{\arg\;\max}\left\{ {b_{i} = 1} \right\}$returns the index of the index of MSB whose b_(i)=1, for example, ifA=001100b, the return value is 3.

In one embodiment, at the output of the first norm approximation module512, the approximated magnitude or approximated normalization term 513successfully replaces the division with a simple right shift. Since theapproximated magnitude is greater than the signal magnitude, the filteris updated with less gain. Without considering the precision issue inthe fixed-point design, the lower gain only implies a slowerconvergence. As the processing is running at tens to hundreds MHz rate,the convergence time difference can be reasonably neglected.

In FIG. 8 , the first update module 535 takes the first filtercoefficient estimate 505 sampled at epoch k−1, the approximatednormalization term 513 from the first norm approximation module 512, thefirst regulated output 525, and the filter output signal 603 (e.g.,error signal output by adder 502) to generate the first coefficientestimate 545 sampled at epoch k. Similarly, the second update module 536takes the second filter coefficient estimate 506 sampled at epoch k−1,the approximated normalization term/output 526 (from the secondregulator 516), and the filter output signal 603 (e.g., error signal) togenerate the second coefficient estimate 546 sampled at epoch k. In oneconfiguration, the algorithm used by the first update module 535 and thesecond update module 536 is modeled as follows or conform to thefollowing equations:

$\begin{matrix}{{a_{i,k} = {a_{i,{k - 1}} + \left\{ {{{reg}_{i,k} \cdot E_{k}} ⪢ b_{approx}} \right\}}},} & {{Equation}\mspace{14mu} 21}\end{matrix}$where

-   -   a_(i,k−1) is the ith coefficient sampled at epoch k−1 (i=1, 2),    -   reg_(i,k) is the ith regulator output sampled at epoch k (i=1,        2),    -   E_(k) is the error signal sampled at epoch k, and    -   b_(approx) is the shifted bits representation of approximated        magnitude, equal to log₂ A_(approx).

As shown by Equation 21, the approximated normalization term (rightshift b_(approx)) is an overestimation of the true magnitude of theerror signal E_(k), i.e., the resultant E_(k)>>b_(approx) is relativelysmaller compared with the exact reference. In terms of the fixed-pointsolution, this does not only imply the long convergence time, but apotential risk of loss in precision. To relieve this potential risk, thedata processor (160, 170) can increase the precision scale to the filtercoefficients and the corresponding path to E_(k) calculation. Todetermine the minimum precision scale while satisfying the convergenceand accuracy requirement, the electronic data processor (160, 170) runsor executes a MonteCarlo (MC) simulation to generate the statisticalmatrix under different scale configurations. In this embodiment thatuses the statistical matrix derived from or generated from theMonteCarlo simulation, the statistics from the simulation shows that agreater number bit scale (e.g., fourteen bits scale) is required ifadopting the approximated normalization approach, compared with a lowernumber bit scale (e.g., eight bits scale) using the exact division.Balancing against the complexity and challenge to the high-speedprocessing by division, the electronic data processor (160, 170) canselect the optimal division processing algorithm based on availableprocessing capacity, through-put or estimated lag time.

FIG. 9 is a block diagram of a fifth embodiment of a cascaded notchfilter 900 based on a mini-batch gradient descent algorithm. In FIG. 9 ,the notch filter 900 provides an example of mini batch update foradaptive learning system. The concept is sometimes mentioned in thecontext of training a neural network, and the concept is very suitablefor the high-speed filtering update. Accordingly, the mini-batch updatecan fully utilize the specialized vector or matrix instruction setprovided by a dedicated digital signal processor, or electronic dataprocessor.

Assuming a mini batch size of q, in FIG. 9 , the feedback update rate ofthe notch filter can be reduced by a factor of q. Instead of updatingthe system parameter based on a filter output signal 603 (e.g., instanterror signal), the update is executed based on an average over the pastq signals. Therefore, the mini-batch update approach not only slows thecomputational rate of the electronic data processors (160, 170), but hasthe potential to fully uses the vector and matrix processing feature ofcertain electronic data processors (160, 170). The mini-batch update fordetermination or filter coefficients also improves the noise resistance.

In FIG. 9 , the filter coefficient vector 656 remains constant over thepast q epochs. For example, the filter coefficient vector 656(Theta_(k−q)) may remain generally constant from k−q+1 to k,corresponding to the mini batch size of q. Instead of using theinstantaneous vector signal 453 and the filter output signal 603 (e.g.,output error signal with a corresponding error component) to drive thefeedback loop, a first electronic memory block 652 is used to store thevector signal 453 sampled in the past q epochs (k−q+1 to k) andresulting a matrix of Equation 22 as follows:

$\begin{matrix}{{\Phi_{v,k} = \begin{pmatrix}\varphi_{k - q + 1} & \ldots & \varphi_{k}\end{pmatrix}},} & {{Equation}\mspace{14mu} 22}\end{matrix}$where

-   -   Φ_(v,k) denotes a matrix of the vector signal 453 (I×1) sampled        over the past q epochs (k−q+1 to k),    -   Φ_(k−q+1) denotes the vector signal 453 (I×1) sampled at epoch        k−q+1, and    -   Φ_(k) is the vector signal 453 (I×1) sampled at epoch k        A second electronic memory block 651 is used to store the filter        output signal 603 (e.g., error signal) sampled in the past q        epochs (k−q+1 to k), which span a vector as E_(v,k)=(E_(k−q+1) .        . . E_(k))^(T). In the notch filter 900 of FIG. 9 , the        covariance matrix update module 657, uses a batch of vector        signal 453, (e.g., which is accumulated to provide or derive        update module input 653) and update module input 654 to update        the covariance in Equation 23 as follows:

$\begin{matrix}{{P_{k} = {P_{k - q} - \frac{{P_{k - q}\Phi_{v,k}\Phi_{v,k}^{H}P_{k - q}^{H}}~}{\Phi_{v,k}^{H}P_{k - q}\Phi_{v,k}}}},} & {{Equation}\mspace{14mu} 23}\end{matrix}$where

-   -   P_(k) is the covariance matrix sampled at epoch k,    -   P_(k−q) is the covariance matrix sampled at epoch k−q,    -   Φ_(v,k) is the set of vector signals sampled from epoch k−q+1 to        k,    -   ( )^(H) denotes the conjugate transpose.        The mini-batch update that is supported by the covariance matrix        update module 657 in the notch filter 900 of FIG. 9 is different        that the update module input 454 of FIG. 7 , for instance. In        the notch filter of FIG. 9 , the coefficients update module 658        uses the mini-batch updated covariance matrix 655 (from        covariance update module 657) and a batch of error signals        sampled from epoch k−q+1 to k to update or determine the filter        coefficients vector in Equation 24 as follows:

$\begin{matrix}{{\overset{\rightharpoonup}{\theta_{k}} = {\overset{\rightharpoonup}{\theta_{k - q}} - \frac{P_{k}\Phi_{\nu,k}E_{\nu,k}^{*}}{q}}},} & {{Equation}\mspace{14mu} 24}\end{matrix}$where

-   -   is the coefficients estimation sampled at epoch k,    -   is the coefficients estimation sampled at epoch k−q,    -   P_(k) is the covariance matrix sampled at epoch k,    -   Φ_(v,k) is the set of vector signals 453 sampled from epoch        k−q+1 to k,    -   E*_(v,k) is the set of filter output signal (203, 503, 603,) or        error signals sampled from epoch k−q+1 to k, and    -   q is the size of the mini batch.

The mini-batch update (e.g., of filter vector coefficients 659 or filtercoefficients) that is supported by the filter coefficients update module658 in the notch filter of FIG. 9 is different that the update moduleinput 454 of FIG. 7 , for instance. The resultant filter coefficientsparameter (e.g., 659) will stay unchanged for the next q epochs, say forexample from k+1 to k+q. Such mini-batch update algorithm also suitablefor or compatible with the simplified notch filter of FIG. 8 , where thefirst update module 535 and second update module 536 can be modified totake the batch of feedback signals to execute the mini batch update.

To measure the signal-to-noise (SNR) degradation resulting from theuncompensated NBI residual in signal (203, 503, 603) a theoreticalanalysis is introduced in this section which provides a very goodreference to predict the notch filter performance.

In FIG. 10 , the vertical axis indicates signal magnitude 751 and thehorizontal axis 752 indicates frequency. The power spectrum density 704of the desired received signal carrier or carriers. The signal 701denotes the desired PN sequence, assuming the PN is centered at “0” withpower of P_(PN) in milliwatts. The baseband bandwidth 702 indicates theaccumulation bandwidth or frequency range of data samples for thedigital baseband processing, for example, 1 KHz bandwidth or upperfrequency limit. The NBI component 705, with power of P_(j) milliwatts,is located at frequency 703 which is ΔF_(J) Hz away from the PN center,such as at 701. The notch depth or peak attenuation 706 can be expressedas a ratio or in decibels (dB). The residual NBI 707 remains afterapplication of the notch filter (e.g., 300, 700, 800, 900). For example,with reference to FIG. 3 , the received signal (115, 135) passes throughthe notch filter system 300, the resultant signal or error signal 341 isdefined in Equation 25 as follows:

$\begin{matrix}{{E = {{PN} + J_{r} + n}},} & {{Equation}\mspace{14mu} 25}\end{matrix}$where

-   -   E denotes the error signal,    -   PN denotes the pseudorandom sequence with power of P_(PN), and    -   J_(r) denotes the residual power of the NBI, and    -   n is the receiver thermal noise.        The power to the residual NBI or interference error component is        as follows in Equation 26:

$\begin{matrix}{{P_{Jr} = {\frac{P_{J}}{D_{NF}}\mspace{14mu}{or}}}{{{d\;{Bm}_{Jr}} = {{dBm_{J}} - {dB_{NF}}}},}} & {{Equation}\mspace{14mu} 26}\end{matrix}$where

-   -   P_(Jr) is the power in milliwatts to NBI component of error        signal 341,    -   P_(J) is power in milliwatt to NBI component of the received        signal or input signal (115, 135, or both),    -   D_(NF) is the notch filter attenuation on NBI in ratio noted by        706,    -   dBm_(Jr) is the power in dBm to NBI component of error signal        341,    -   dBm_(J) is the power in dBm to NBI component of the received        signal or input signal (115, 135, or both),    -   dB_(NF) is the notch filter attenuation on NBI in dB noted by        706.        The error signal 341 is despreaded by the local PN replica.        Assuming the perfect alignment with the PN component in error        signal 341, the de-spreading process concentrates the power of        PN into frequencies from “0” to 702; it also spreads the line        spectrum of residual NBI 707 into a power spectrum density 704        with generally sinc pattern as set forth in Equation 27, and as        illustrated in FIG. 10 .

$\begin{matrix}{{{PS{D_{Jr}(f)}} = {{P_{Jr} \cdot \sin}\;{c^{2}\left( {f - {\Delta F_{J}}} \right)}}},} & {{Equation}\mspace{14mu} 27}\end{matrix}$where

-   -   PSD_(Jr)(f) denotes the power spectrum density (704) of the NBI        despreaded by a PN sequence,    -   P_(Jr) is the power of the residual NBI (707) contained in the        error signal 341,    -   sinc(x) is the sinc function defined as sin(πx)/(πx), and    -   ΔF_(J) denotes the frequency separation between the NBI located        at frequency 703 and the PN sequence located at “0”.        The integration of the sinc pattern over whole frequency range        is 1 in Equation 28 as follows:

$\begin{matrix}{1 = {\int_{- \infty}^{+ \infty}{\sin\;{c^{2}\left( {f - {\Delta F_{J}}} \right)}df}}} & {{Equation}\mspace{14mu} 28}\end{matrix}$In the frequency response 750 of FIG. 10 , the bandwidth betweenreference frequency pairs (733, 713; 713, 703; 703, 723; 723, 743) alongthe frequency axis 752 are all equal to the chipping rate of PN signal.Integrating the power spectrum density 704 from within the basebandbandwidth 702 (e.g., from 0 Hz to a frequency limit at the upper orrightmost vertical edge of the baseband bandwidth 702) estimates thepower of residual NBI that leaks into the demodulator output, noted as708 as follows:

$\begin{matrix}{{J_{r0} = {\frac{1}{F}{\int_{0}^{F}{PS{D_{Jr}(f)}df}}}},} & {{Equation}\mspace{14mu} 29}\end{matrix}$where

-   -   PSD_(Jr)(f) is the power spectrum density of 704,    -   F is the bandwidth of demodulator output, 702, and    -   J_(r0) is the averaged power spectrum density within bandwidth        of F.        Based on Equation 29, the SNR degradation can be computed in        Equation 30 as follows:

$\begin{matrix}{{L = {1 + \frac{J_{r\; 0}}{N_{0}}}},} & {{Equation}\mspace{14mu} 30}\end{matrix}$where

-   -   J_(r0) is the averaged power spectrum density of NBI within the        bandwidth of F,    -   N₀ is the power spectrum density of the thermal noise, and    -   L is the SNR degradation in ratio.

A receiver with an illustrative adaptive notch filter system (e.g., 300,700, 800, 900) is configured to evaluate possible attenuationperformance consistent with a modeled example in FIG. 11 through 15 ,inclusive.

FIG. 11 is an illustrative example of a chart of magnitude versusfrequency response of the adaptive notch filter, where an adaptivesolution and an expected or benchmark solution are compared. The signalmagnitude 805 is illustrated on the vertical axis and the frequency isillustrated on the horizontal axis 803. The notch filter response has adual notch response comprises a first notch 802 and a second notch 804that are separated by a frequency separation 809, where the second notch804 is aligned with a second notch center frequency 808 that attenuatesa NBI component 812. Meanwhile, the first notch 802 has a first notchcenter frequency 806. The reference magnitude 807 is aligned with aconstant or uniform magnitude (or a horizontal line representative) ofzero decibels along the axis of the signal magnitude 805

The model parameters of the modeled adaptive notch filter are set forthin Table 3 for simulation setup for the performance evaluation of theillustrative notch filter. For example, the system has ananalog-to-digital converter (ADC) of 8 bits and filter coefficientsscale of 14 bits for the associated, modeled adaptive notch filter. Itsgoal is to attenuate two continuous wave (CW) interference signalcomponents (J1/N and J2/N) that are 42 dB above the total noise power ornoise floor, where the frequency of the first NBI component (F_(J1)) isat 5 MHz and where the frequency of the second NB component (F_(J2)) isat 15 MHz. The pseudorandom noise code (PNr) modulation as at afrequency of 1.023 MHz and is signal to noise ratio (C/NO) is 45decibels with respect to the noise floor.

TABLE 3 Simulation/Modeled Configuration PNr C/N0 F_(J1) J1/N F_(J2)J2/N ADC Coefficient (MHz) (PN) (MHz) (dB) (MHz) (dB) (bits) Scale(bits) 1.023 45 5 42 15 42 8 14Table 4 in conjunction with FIG. 11 illustrate the frequency response ofthe modeled adaptive notch filter versus the ideal frequency response ofan ideal notch filter, where the ideal frequency response providesadditional attenuation at the notch frequency as best illustrated by thedashed lines in rightmost notch of FIG. 11 .

TABLE 4 Frequency Response of the Ideal Notch Filter vs. the ModeledAdaptive Notch Filter at Steady State (FIG. 810) Magnitude Response Ref.No. D(FJ1) dB D(FJ2) dB Ideal Notch Filter 812 −42 −42 Adaptive NotchFilter @ 802, 804 −43 −41 1 ms convergenceFurther, FIG. 11 demonstrates the frequency response between the idealnotch filter (based on the predefined or a priori knowledge of therespective frequencies of the two NBI components) and the adaptivefrequency response of system 500 after a convergence period (e.g.,lmillisecond (ms) running of the filter after initialization). Table 4lists the notation and notch depth in FIG. 11 , which implies theadaptive frequency response of system 500 converges to the ideal in ⅛ ofthe 1 ms˜=125 uS.

TABLE 5 Coefficients of Adaptive Notch Filter vs. the DerivedCoefficients of Ideal Notch Filter Ref Adaptive Filter CoefficientCoefficients No. (Value) @ 1 ms Convergence TRUE A0.Re 821 −0.05 −0.05(Real) A0.Im 822   1.44   1.438 (Imaginary) A1.Re 823 −1 −0.99 (Real)A1.IM 824 −0.06 −0.07 (Imaginary)

FIG. 12 is an illustrative example of a chart that shows convergenceversus time of the filter coefficients to a steady-state solution offilter coefficients as a reference. In FIG. 12 the vertical axis showsthe filter coefficients 828 and the horizontal axis 827 shows time.

Table 5 in conjunction with FIG. 12 demonstrate the convergence processof the filter coefficients in a series of filter coefficient curves 820.The vertical axis represents the filter coefficients axis 828, whereasthe horizontal axis represents time 827. Here, in the example, thefilter coefficients are initialized with “0”, and average anillustrative convergence period (e.g., at about 125 uS), the filtercoefficients all converge very close to the expected value as shown byTable 5. In FIG. 12 , each filter coefficient (AO, A1) has a realcomponent and an imaginary component. For example, the first filtercoefficient (A0) has a real filter coefficient (AO.RE) indicated byreference number 821, which is illustrated as a dashed reference line,that reaches convergence at inflection point 829; the first filtercoefficient (A0) has an imaginary filter coefficient (AO.Im) indicatedby reference number 822, which is illustrated as a alternating dot-dashreference line, that reaches convergence at inflection point 830; thesecond filter coefficient (A0) has a real filter coefficient (A1.RE)indicated by reference number 823, which is illustrated as a solidline/curve, that has an inflection point at 831; the second filtercoefficient (A1) has an imaginary filter coefficient (A1.IM) indicatedby reference number 824, which is illustrated as a short dashedreference line; that has an inflection point at 832.

FIG. 13 compares the group delay between the group delay of ideal NF andthe ANF. It demonstrates the system 500 can converge to the expectationvery well. The notations are listed in Table 5.

TABLE 6 Group Delay of Ideal Notch Filter vs. Adaptive Notch Filter atSteady State Group Delay Ref. No. Ideal Notch Filter 831 Adaptive NotchFilter @ 1 ms 832 convergence

FIG. 13 is an illustrative example of a chart 850 that shows group delayof the notch filter over an extended bandwidth of potential interest. InFIG. 13 , the vertical axis shows group delay or delay in samples orsampling intervals, whereas the horizontal axis shows frequency (e.g.,in MHz). Table 5 in conjunction with FIG. 13 , illustrates the groupdelay in samples or sampling interval versus frequency of the adaptivenotch filter. Group delay means the characteristic of phase change orphase distortion per unit frequency for transmission through theadaptive notch filter. If the group delay exceeds a threshold, thereceived radio frequency signal and the baseband signal may bedistorted, such as a change in the envelope of the wave, which canimpact demodulation of the pseudorandom noise code or other modulationof the radio frequency signal.

In FIG. 13 , the vertical axis represents the delay in samples 888 orsampling interval, whereas the horizontal axis represents frequency 889(e.g., in MHz). The illustrative dashed lower curve 832 with less groupdelay is indicated by reference curve 832. The illustrative dashed lowercurve 832 is an expected curve for group delay that is a practicaltarget value or design goal of group delay for the adaptive notchfilter. In contrast, an ideal group delay response (825, 834) has nodelay difference associated with a corresponding change in frequency,although such ideal group delay response is not attainable in practice.The solid upper curve 833 with greater group delay is indicated byreference curve 833 and the modeled performance will generally fallbetween the dashed lower curve 832 and the solid upper curve 833.

FIG. 14 is an illustrative example of graph of potential filter error ina magnitude of the desired, received signal compared betweenpre-convergence and post-convergence. FIG. 14 illustrate the convergenceof error signal 841 (e.g., same or similar to one or more of (error)signals 203, 341, 503, and/or 603). In FIG. 14 , the vertical axisrepresents signal magnitude 891 and the horizontal-axis represents time892. Accordingly, in FIG. 14 the error signal has a real component(E.Re) and imaginary component (E.Im), where the imaginary component 842falls within a region bounded by the long-dashed line and where the realcomponent 841 coincides with curve 841, which is illustrated inshort-dashed lines.

In FIG. 14 , before the adaptive notch filter (ANF) converges on filtercoefficients (e.g., filter vector coefficients) to attenuate the NBIcomponents, the magnitude of the error signal is close to that of thereceived signal or input signal (e.g., 115, 135, or both) and the NBIcomponents 852 can dominate the power measurement. However, after theadaptive notch filter (ANF) converges after a convergence time period(e.g., at the epoch associated with the lapse of 125 uS frominitialization of the ANF), the energy of the error signal 341 canreduce to a very small value, which tends to indicate the NBI componentsare significantly attenuated. FIG. 14 in conjunction with Table 7specifies the following:

TABLE 7 The Waveform of Error Signal 341 before vs. after ConvergenceError Signal 341 Ref. No. E.Re (Real) 841 E.Im (Imaginary) 842

FIG. 15 is illustrative example of histogram of the desired, receivedsignal versus an error signal at a steady state. In FIG. 15 , thevertical axis represents the number of occurrences (count) 854 of areceived signal component of a certain magnitude, whereas the horizontalaxis represents frequency 853. FIG. 15 demonstrates the energy change,in terms of the histogram, of the error signal 341, before versus afterconvergence. The strong NBI components tend to cause the histogram ofthe received signal or input signal (e.g., 115, 135 or both) to bewidely spread out as shown by the histogram curve 851 prior toconvergence of the active notch filter, such as illustrated by the solidlines, solid arcs or solid curves. In contrast, after the convergence ofthe adaptive notch filter to steady-state, the histogram curve 852 orresponse of the error signal (e.g., 341) after ANF becomes narrowlyfocused over a narrow bandwidth, as indicated by or similar to thedashed lines. FIG. 15 can be interpreted together with Table 8 asfollows:

TABLE 8 The Histogram Comparison, Received Signal 115/135 vs. ErrorSignal 341 Histogram Ref. No. Signal 115/135 851 Signal 341 852

In the above table and through the document, “115/135” means signal 115,signal 135 or both signals 115 and 135. The adaptive notch filter andmethod discloses a general model to unify bothinfinite-impulse-response-based notch filter (NF) and afinite-impulse-response-based notch filter. In other words, the adaptivenotch filter is well-suited for operation in the infinite impulseresponse (IIR) mode, the finite impulse response (FIR) mode, or thehybrid or dual mode filter may be referred to as a transverse filter.For example, the adaptive notch filter facilitates a physicallyrealizable IIR based notch-filter (NF) using a Steiglitz-McBride (SM)method or a quasi-SM (QSM) method where QSM estimator modes mayfluctuate over time within or more of the following modes: SM mode, LSMmode, and/or MMSE mode. Further, the SM mode pay be applied to the IIRmode, whereas the LSM mode and the MMSE mode may be well-suited foroperation in the FIR mode.

The adaptive notch filter is well suited to define and evaluate thebandwidth (BW) for the first order notch filter and provides the visualillustration between the bandwidth and tunable parameters. In theadaptive notch filter, a second signal processing path (e.g., second setof delay lines with corresponding taps), against the first signalprocessing path (e.g., first set of delay lines with correspondingtapes), can be delayed by arbitrary number of samples to guarantee thedecorrelation property to the first path (e.g., to facilitate rapidconvergence or estimation by existing, commercially available electronicdata processors). In one configuration, a Monte-Carlo (MC) simulationcan be used to demonstrate that, without considering the correlationbetween multiple narrowband interference (NBI) components, the adaptivenotch filter can reliably reject up to two NBI components. Further, thesimplified design minimizes the computation delay and is suitable forhigh-speed processing (e.g., multiple bits system at tens to hundreds ofMHz rate).

In this disclosure certain configurations of the adaptive notch filtersupports features such as one or more of the following, to the extentapplicable for the embodiment or configuration: (a) a cascaded structurewhich can maintains the length of parameter vector the same as basicnotch filter system while improving the attenuation by approximatelyN×30 dB; (b) a pre-scale to the pre-emphasis filtering or adaptive lineenhancer (ALE) to reduce possibly the logic size of the digital systemwhile balancing an acceptable performance; (c) avoid the covariancematrix propagation by disregarding the correlation between multipleinterferences; (d) potentially improves performance by adding additionalbits on the multiplication path to compensate the precision loss by thedivision approximation; (e) uses a mini batch gradient-descent approachto benefit potentially from the vector processing unit which, comparedwith the stochastic gradient-descent method, can improve the noiseresistance and improves the reliability; and (f). introduces thequantity analysis measuring the signal to noise ratio degradationrelated with the power and frequency separation between the narrowbandinterference and pseudorandom liked signal; and (g) a unit gainregulator to the ALE outputs sampled at previous epochs of a satellitenavigation receiver.

In one embodiment, the pre-scaling, the pre-emphasis filtering or ALEconfigures a wide-bit multiplier (e.g., at least 30 bit A times 30 bitB) to address any significant timing constraint in the a specificintegrated circuit (ASIC) design that would otherwise make it difficultfor realize high speed data processing and real-time operation of theadaptive filter (e.g., by existing commercially available dataprocessors at reasonable product or service offering cost).

In one embodiment, the adaptive notch filter comprises a unit gainregulator to the ALE outputs sampled at previous I epochs. To estimatethe filter coefficients, the ALE samples at epoch k, which is a linearcombination of the ALE outputs sampled from epoch k−1 to k−I. The ALE ordata processing multiples a conjugate of the error signal sampled atepoch k by the ALE outputs sampled at k−m (m=1 . . . I) to generate theupdate signal to adjust the mth coefficient. Before updating the mthcoefficient, a normalizer or electronic data processor normalizes themth update signal, where the mth update signal is normalized by theamplitude of ALE output sampled at k−m and by the amplitude of the errorsignal sampled at epoch k. The normalizer or unit gain regulator mayfacilitate any of the following: elimination the I large complexmultipliers, simplification of the normalization, and reduction ofdisturbance from the noise of the ALE outputs.

In one embodiment, the adaptive notch filter introduces a new amplitudenormalization method. To normalize the data for filter coefficientestimation, the electronic data processor generally uses a divisionfunction that requires multiple clock cycles to complete. For example,if the electronic data processor is processing the data at X MHz and thedivision requires Y clock cycles to complete the calculation. Thedivision logic needs to run at (X multiplied by Y) MHz to make thereal-time processing possible. To further simplify the normalizationcalculation, the electronic data processor may use a round-up functionto round the amplitude up to a certain level to converting the divisioninto a simple shift register operation (e.g., right shift); hence,making normalization determinations potentially compatible withhigh-speed processing in a wide-bit system.

Although certain embodiments of receivers, systems, methods, processesand examples have been described in this disclosure, the scope of thecoverage of this disclosure may extend to variants of the receiver,systems, methods, processes and examples and systems and conceptsdisclosed herein. For example, in any patent that may be granted on thisdisclosure, one or more claims can cover equivalents and variants to thefull extent permitted under applicable law, among other things

The following is claimed:
 1. A receiver system with interferencerejection, the receiver system comprising: an antenna for receiving aradio frequency signal; a downconverter for converting the radiofrequency signal to an intermediate frequency signal; ananalog-to-digital converter for converting the intermediate frequencysignal or an analog baseband signal to a digital baseband signal; aselective filtering module for filtering of the digital baseband signalconsistent with a target receive bandwidth; a narrow band rejectionfilter for rejecting an interference component, wherein the selectivefiltering module comprises: an adaptive notch filter supporting aninfinite impulse response mode; and a controller for controlling theadaptive notch filter, the controller configured to execute aSteiglitz-McBride (SM)-modeled estimator or a least mean squaresalgorithm estimator to estimate filter coefficients of the adaptivenotch filter, or the controller configured to execute cumulatively boththe Steiglitz-McBride (SM)-modeled estimator and the least mean squares(LMS) algorithm estimator, the SM-modeled estimator or least meansquares algorithm estimator being storable in a data storage device andthe controller configured to recursively adjust the filter coefficientsof the adaptive notch based on the estimated filter coefficients;wherein the SM-modeled estimator or the LMS estimator is configured toconduct a mini batch gradient decent search, wherein the SM-modeledestimator or LMS estimator or a filter coefficient updater is configuredto avoid a covariance matrix update/propagation by disregardingcorrelation between two NBI components for a scenario or case whereinthere are two simultaneous NBI components that interfere with thereceived radio frequency signal or its related respective basebandsignal that is derived from the received radio frequency signal.
 2. Thereceiver according to claim 1 wherein the SM-modeled estimator or theLMS estimator is configured to reduce a feedback update rate by a factorof q, where q is batch size.
 3. The receiver according to claim 1wherein the SM-modeled estimator or the LMS estimator is configured touse a separate covariance matrix update and coefficient matrix update.4. The receiver according to claim 1 wherein the SM-modeled estimator orthe LMS estimator is configured to support use of specialized vector ormatrix instruction set associated with a corresponding data processor.5. The receiver according to claim 1 wherein the SM-modeled estimator orthe LMS estimator is configured to search or conduct a mini batchgradient decent search among candidate filter coefficients for refiningfilter coefficients consistent with the following equations, where afilter coefficient can remain generally constant over the past q epochs(k−q+1 to k): ${\Phi_{v,k} = \begin{pmatrix}\varphi_{k - q + 1} & \ldots & \varphi_{k}\end{pmatrix}},$ where Φ_(v,k) denotes a matrix of the vector signal 453(I×1) sampled over the past q epochs (k−q+1 to k), φ_(k−q+1) denotes thevector signal 453 (I×1) sampled at epoch k−q+1, and φ_(k) is the vectorsignal 453 (I×1) sampled at epoch k.
 6. The receiver according to claim5 wherein in the adaptive notch filter comprising a covariance matrixupdate module, and wherein the covariance matrix update module uses abatch of vector signals to update the covariance as follows in the belowequation:${P_{k} = {P_{k - q} - \frac{P_{k - q}\Phi_{v,k}\Phi_{v,k}^{H}P_{k - q}^{H}}{\Phi_{v,k}^{H}P_{k - q}\Phi_{v,k}}}},$where P_(k) is the covariance matrix sampled at epoch k, P_(k−q) is thecovariance matrix sampled at epoch k−q, Φ_(v,k) is the set of vectorsignals sampled from epoch k−q+1 to k, ( )^(H) denotes the conjugatetranspose.
 7. The receiver according to claim 5 wherein in the adaptivenotch filter comprising a coefficients update module, and wherein thecoefficients update module uses the mini-batch updated covariance matrixand a batch of error signals sampled from epoch k−q+1 to k to update thecoefficients vector as follows in the below equation:${\overset{\rightharpoonup}{\theta_{k}} = {\overset{\rightharpoonup}{\theta_{k - q}} - \frac{P_{k}\Phi_{\nu,k}E_{\nu,k}^{*}}{q}}},$where

is the coefficients estimation sampled at epoch k,

is the coefficients estimation sampled at epoch k−q, P_(k) is thecovariance matrix sampled at epoch k, Φ_(v,k) is the set of vectorsignals sampled from epoch k−q+1 to k, E*_(v,k) is the set of outputfilter signals or error signals sampled from epoch k−q+1 to k, and q isthe size of the mini batch.
 8. The receiver according to claim 1 whereinthe mini batch gradient descent improves the noise resistance andimproves the reliability with respect to a stochastic gradient-descentmethod.
 9. The receiver system according to claim 1 wherein theselective filter module further comprises: a secondary adaptive notchfilter, wherein the adaptive notch filter attenuates a received signalat first notch frequency in a first frequency band and wherein thesecondary adaptive notch filter attenuates the received signal at asecond notch frequency in a second frequency band.
 10. A receiver systemwith interference rejection, the receiver system comprising: an antennafor receiving a radio frequency signal; a downconverter for convertingthe radio frequency signal to an intermediate frequency signal; ananalog-to-digital converter for converting the intermediate frequencysignal or an analog baseband signal to a digital baseband signal; aselective filtering module for filtering of the digital baseband signalconsistent with a target receive bandwidth; a narrow band rejectionfilter for rejecting an interference component, wherein the selectivefiltering module comprises: an adaptive notch filter supporting aninfinite impulse response mode; and a controller for controlling theadaptive notch filter, the controller configured to execute aSteiglitz-McBride (SM)-modeled estimator or a least mean squaresalgorithm estimator to estimate filter coefficients of the adaptivenotch filter, or the controller configured to execute cumulatively boththe Steiglitz-McBride (SM)-modeled estimator and the least mean squares(LMS) algorithm estimator, the SM-modeled estimator or least meansquares algorithm estimator being storable in a data storage device andthe controller configured to recursively adjust the filter coefficientsof the adaptive notch based on the estimated filter coefficients;wherein the notch filter is physically realized as an infinite impulseresponse filter that uses an artificial intelligence (AI) model toconverge on filter coefficients more quickly than a least minimumsquares technique, wherein the filter coefficients are determined basedon a mini-batch stochastic gradient descent search.
 11. The receiversystem according to claim 10 wherein the controller is configured todefine a bandwidth for a first order notch filter about a target notchfrequency, wherein the bandwidth is wider than a minimum bandwidthrequired to reject an interfering carrier signal modulated with phaseshift keying or bipolar phase shift keying.
 12. The receiver systemaccording to claim 10 wherein the AI model comprises a SM-modeledalgorithm.
 13. The system according to claim 10 wherein the stochasticgradient search technique satisfies a Steiglitz-McBride criteria. 14.The receiver system according to claim 10 wherein the notch filter isphysically realized as an infinite impulse response filter that uses aSteiglitz-McBride technique (or algorithm) to determine or convergeiteratively on selected filter coefficients from a plurality ofcandidate filter coefficients.
 15. The receiver system according toclaim 10 wherein the filter coefficients conform to the followingequation:${{{A\left( {\rho z} \right)} = {\prod_{i = 1}^{I}\left( {1 - {\rho\beta_{i}e^{j\omega_{i}}z^{- 1}}} \right)}}{= {1 + {\sum_{i = 1}^{I}{a_{i}^{*}\rho^{i}z^{- i}}}}}},$where A(ρz) defines the frequency (or frequencies) to be attenuated,such as a target notch frequency or target notch frequencies (e.g.,target magnitude versus frequency response) of the adaptive notchfilter; ρ is a filter parameter, which in combination with filtercoefficient β_(i) can be used to control the bandwidth and depth of thenotch to each narrowband interference component (NBI), Π_(i=1)^(I)(1−β_(i)e^(jω) ^(i) z⁻¹) is a multiplication representation of a setof NBI component from a first NBI to an ith NBI up to a total of I NBIcomponents; I indicates the total number of the narrow band interference(NBI component) that need to be rejected or notched, or the order of theadaptive notch filter or the degree of freedom (D.O.F); e^(jω) ^(i) isthe ith NBI modeled as a simple continuous wave (CW); β_(i) is a filtercoefficient of the adaptive notch filter that determines a notch depthto the ith NBI component and in range of [0, 1]; z^(−i) represents atime delay or delay line (e.g., in the z-transform domain) associatedwith the ith tap of the adaptive narrow band filter; 1+Σ_(i=1)^(I)a*_(i)z^(−i) is a sum representation, equivalent to themultiplication representation Π_(i=1) ^(I)(1−β_(i)e^(jω) ^(i) z⁻¹),where a*_(i) is the conjugate complex to a_(i); a_(i) is a filtercoefficient of the adaptive notch filter; and 1+Σ_(i=1) ^(I)a*_(i)z^(−i)is the sum representation of denominator; this term cannot be reducedfor the IIR based adaptive notch filter.
 16. The receiver systemaccording to claim 10 wherein the notch filter comprises a cascadedinfinite impulse response structure that can adapt to the notch depthand meet the limitations/specifications of the analog-to-digitalconverter.
 17. The receiver system according to claim 16 wherein thecascaded infinite impulse response structure comprises a filtercoefficients to create a multiple stage adaptive notch filter consistentwith the below equation as follows: H_(N)(e^(jω₁)) = H^(N)(e^(jω₁)),where N is the number of stages; and e^(jω) ^(i) is the ith NBI modeledas a simple continuous wave (CW), where for the first order filter modeli and I equals
 1. 18. The receiver system according to claim 10 whereinpost-convergence of the Steiglitz-McBride technique, the notch thecontroller is configured to execute a least mean squares technique(algorithm) to recursively adjust the coefficients of the adaptive notchfilter to adapt a notch frequency in real time to mitigate theinterference component impact on a pseudo-noise (PN) sequencedemodulation of the radio frequency signal.
 19. The receiver systemaccording to claim 10 wherein the selective filter module furthercomprises: a secondary adaptive notch filter, wherein the adaptive notchfilter attenuates a received signal at first notch frequency in a firstfrequency band and wherein the secondary adaptive notch filterattenuates the received signal at a second notch frequency in a secondfrequency band.